{"id":1136,"date":"2018-12-11T13:23:16","date_gmt":"2018-12-11T18:23:16","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/use-a-general-strategy-to-solve-linear-equations-2\/"},"modified":"2018-12-11T13:23:16","modified_gmt":"2018-12-11T18:23:16","slug":"use-a-general-strategy-to-solve-linear-equations-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/use-a-general-strategy-to-solve-linear-equations-2\/","title":{"raw":"Use a General Strategy to Solve Linear Equations","rendered":"Use a General Strategy to Solve Linear Equations"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Solve linear equations using a general strategy<\/li><li>Classify equations<\/li><li>Solve equations with fraction or decimal coefficients<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167825703510\" class=\"be-prepared\"><p id=\"fs-id1167833364840\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167833387029\" type=\"1\"><li>Simplify: \\(\\frac{3}{2}\\left(12x+20\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829788421\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(5-2\\left(n+1\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167836503089\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Find the LCD of \\(\\frac{5}{6}\\) and \\(\\frac{1}{4}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836518722\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836663302\"><h3 data-type=\"title\">Solve Linear Equations Using a General Strategy<\/h3><p id=\"fs-id1167826205047\">Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that makes it a true statement. Any value of the variable that makes the equation true is called a <span data-type=\"term\">solution<\/span> to the equation. It is the answer to the puzzle!<\/p><div data-type=\"note\" id=\"fs-id1167836341048\"><div data-type=\"title\">Solution of an Equation<\/div><p id=\"fs-id1167833186674\">A <strong data-effect=\"bold\">solution<\/strong> of an equation is a value of a variable that makes a true statement when substituted into the equation.<\/p><\/div><p id=\"fs-id1167829634269\">To determine whether a number is a solution to an equation, we substitute the value for the variable in the equation. If the resulting equation is a true statement, then the number is a solution of the equation.<\/p><div data-type=\"note\" id=\"fs-id1167836487400\" class=\"howto\"><div data-type=\"title\">Determine Whether a Number is a Solution to an Equation.<\/div><ol id=\"fs-id1167836305814\" type=\"1\" class=\"stepwise\"><li>Substitute the number for the variable in the equation.<\/li><li>Simplify the expressions on both sides of the equation.<\/li><li>Determine whether the resulting equation is true. <ul id=\"fs-id1167829788831\" data-bullet-style=\"bullet\"><li>If it is true, the number is a solution.<\/li><li>If it is not true, the number is not a solution.<\/li><\/ul><\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167836388234\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836514463\"><div data-type=\"problem\" id=\"fs-id1167836623770\"><p id=\"fs-id1167833255992\">Determine whether the values are solutions to the equation: \\(5y+3=10y-4.\\)<\/p><p id=\"fs-id1167836596040\"><span class=\"token\">\u24d0<\/span>\\(y=\\frac{3}{5}\\)<span class=\"token\">\u24d1<\/span>\\(y=\\frac{7}{5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829851798\"><p id=\"fs-id1167836610243\">Since a solution to an equation is a value of the variable that makes the equation true, begin by substituting the value of the solution for the variable.<\/p><p id=\"fs-id1167829720215\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829742512\" class=\"unnumbered unstyled\" summary=\"5 y plus 3 is equal to 10 y minus 4. Substitute three-fifths for y. Is the product of 5 and three-fifths plus 3 equal to the product of 10 and three-fifths minus 4. Multiply on each side of the equation. Is 3 plus 3 equal to 6 minus 4? Simplify on each side. 6 is not equal to 2. Since y is equal to three-fifths does not result in a true equation, y is equal to three-fifths is not a solution to the equation 5 y plus 3 is equal to 10 y minus 4.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836791181\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829811131\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836599202\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829691775\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981020\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836433419\">Since \\(y=\\frac{3}{5}\\) does not result in a true equation, \\(y=\\frac{3}{5}\\) is not a solution to the equation \\(5y+3=10y-4.\\)<\/p><p id=\"fs-id1167829714257\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836539230\" class=\"unnumbered unstyled\" summary=\"Substitute seven-fifths for y. Is the product of 5 and seven-fifths plus 3 equal to the product of 10 and seven-fifths minus 4. Multiply on each side of the equation. Is 7 plus 3 equal to 14 minus 4? Simplify on each side. 10 is equal to 10. Since y is equal to seven-fifths results in a true equation, y is equal to seven-fifths is a solution to the equation 5 y plus 3 is equal to 10 y minus 4.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836572094\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836320934\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829894394\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833058014\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829624689\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836352765\">Since \\(y=\\frac{7}{5}\\) results in a true equation, \\(y=\\frac{7}{5}\\) is a solution to the equation \\(5y+3=10y-4.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836542370\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836492374\"><div data-type=\"problem\" id=\"fs-id1167836310213\"><p id=\"fs-id1167836621135\">Determine whether the values are solutions to the equation: \\(9y+2=6y+3.\\)<\/p><p id=\"fs-id1167833036727\"><span class=\"token\">\u24d0<\/span>\\(y=\\frac{4}{3}\\)<span class=\"token\">\u24d1<\/span>\\(y=\\frac{1}{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836692114\"><p id=\"fs-id1167826131759\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833008189\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836579562\"><div data-type=\"problem\" id=\"fs-id1167833339605\"><p id=\"fs-id1167833058165\">Determine whether the values are solutions to the equation: \\(4x-2=2x+1.\\)<\/p><p id=\"fs-id1167829619789\"><span class=\"token\">\u24d0<\/span>\\(x=\\frac{3}{2}\\)<span class=\"token\">\u24d1<\/span>\\(x=-\\frac{1}{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833347056\"><p id=\"fs-id1167836507257\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p><\/div><\/div><\/div><p id=\"fs-id1167836335358\">There are many types of equations that we will learn to solve. In this section we will focus on a <span data-type=\"term\">linear equation<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167836515647\"><div data-type=\"title\">Linear Equation<\/div><p>A <strong data-effect=\"bold\">linear equation<\/strong> is an equation in one variable that can be written, where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers and \\(a\\ne 0,\\) as:<\/p><div data-type=\"equation\" id=\"fs-id1167829644900\" class=\"unnumbered\" data-label=\"\">\\(ax+b=0\\)<\/div><\/div><p id=\"fs-id1167833050177\">To solve a linear equation it is a good idea to have an overall strategy that can be used to solve any linear equation. In the next example, we will give the steps of a general strategy for solving any linear equation. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.<\/p><div data-type=\"example\" id=\"fs-id1167836432956\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Linear Equation Using a General Strategy<\/div><div data-type=\"exercise\" id=\"fs-id1167836477553\"><div data-type=\"problem\" id=\"fs-id1167836333480\"><p id=\"fs-id1167836700210\">Solve: \\(7\\left(n-3\\right)-8=-15\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836294565\"><span data-type=\"media\" id=\"fs-id1167833328021\" data-alt=\"Step 1 is to simplify each side of the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Use the Distributive Property. The equation first simplifies to 7 n minus 21 minus 8 is equal to negative 15. Then it simplifies to 7 n minus 29 is equal to negative 15. Notice that each side of the equation is now simplified as much as possible.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to simplify each side of the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Use the Distributive Property. The equation first simplifies to 7 n minus 21 minus 8 is equal to negative 15. Then it simplifies to 7 n minus 29 is equal to negative 15. Notice that each side of the equation is now simplified as much as possible.\"><\/span><span data-type=\"media\" id=\"fs-id1167836683590\" data-alt=\"Step 2 is to collect all variable terms on the left side of the equation, 7 n minus 29 is equal to negative 15. Notice there is nothing to do because all n\u2019s are on the left side.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to collect all variable terms on the left side of the equation, 7 n minus 29 is equal to negative 15. Notice there is nothing to do because all n\u2019s are on the left side.\"><\/span><span data-type=\"media\" id=\"fs-id1167829712528\" data-alt=\"Step 3 is to collect all constant terms on the other side of the equation, 7 n minus 29 is equal to negative 15. To get constants only on the right, add 29 to each side. The result is 7 n minus 29 plus 29 is equal to negative 15 plus 29. Simplify. The result is 7 n is equal to 14.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to collect all constant terms on the other side of the equation, 7 n minus 29 is equal to negative 15. To get constants only on the right, add 29 to each side. The result is 7 n minus 29 plus 29 is equal to negative 15 plus 29. Simplify. The result is 7 n is equal to 14.\"><\/span><span data-type=\"media\" id=\"fs-id1167829749807\" data-alt=\"Step 4 is to make the coefficient of the equation, 7 n is equal to 14, 1. Divide each side of the equation by 7. The result is 7 n divided by 7 is equal to 14 divided by 7. Simplify. The result is n is equal to 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to make the coefficient of the equation, 7 n is equal to 14, 1. Divide each side of the equation by 7. The result is 7 n divided by 7 is equal to 14 divided by 7. Simplify. The result is n is equal to 2.\"><\/span><span data-type=\"media\" id=\"fs-id1167836615156\" data-alt=\"Step 5 is to check the solution, n is equal to 2, by substituting into the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Is the product of 7 and the quantity 2 minus 3 minus 8 equal to negative 15? Subtract. Is 7 times negative 1 minus 8 equal to negative 15? Is negative 7 minus 8 equal to negative 15. Negative 15 is equal to negative 15. The solution checks.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003e_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to check the solution, n is equal to 2, by substituting into the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Is the product of 7 and the quantity 2 minus 3 minus 8 equal to negative 15? Subtract. Is 7 times negative 1 minus 8 equal to negative 15? Is negative 7 minus 8 equal to negative 15. Negative 15 is equal to negative 15. The solution checks.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836389216\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836613031\"><div data-type=\"problem\" id=\"fs-id1167829694670\"><p id=\"fs-id1167836393244\">Solve: \\(2\\left(m-4\\right)+3=-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836628710\"><p id=\"fs-id1167836605160\">\\(m=2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829739303\"><div data-type=\"problem\" id=\"fs-id1167836599620\"><p id=\"fs-id1167836375852\">Solve: \\(5\\left(a-3\\right)+5=-10.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836512901\"><p id=\"fs-id1167836608138\">\\(a=0\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167829683897\">These steps are summarized in the <span data-type=\"term\" class=\"no-emphasis\">General Strategy for Solving Linear Equations<\/span> below.<\/p><div data-type=\"note\" id=\"fs-id1167836530302\" class=\"howto\"><div data-type=\"title\">Solve linear equations using a general strategy.<\/div><ol id=\"fs-id1167836375366\" type=\"1\" class=\"stepwise\"><li>Simplify each side of the equation as much as possible.<div data-type=\"newline\"><br><\/div> Use the Distributive Property to remove any parentheses.<div data-type=\"newline\"><br><\/div> Combine like terms.<\/li><li>Collect all the variable terms on one side of the equation.<div data-type=\"newline\"><br><\/div> Use the Addition or Subtraction Property of Equality.<\/li><li>Collect all the constant terms on the other side of the equation.<div data-type=\"newline\"><br><\/div> Use the Addition or Subtraction Property of Equality.<\/li><li>Make the coefficient of the variable term equal to 1.<div data-type=\"newline\"><br><\/div> Use the Multiplication or Division Property of Equality.<div data-type=\"newline\"><br><\/div> State the solution to the equation.<\/li><li>Check the solution.<div data-type=\"newline\"><br><\/div> Substitute the solution into the original equation to make sure the result is a true statement.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167836481206\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836389542\"><div data-type=\"problem\" id=\"fs-id1167829597819\"><p id=\"fs-id1167832930220\">Solve: \\(\\frac{2}{3}\\left(3m-6\\right)=5-m.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836424231\"><table id=\"fs-id1167836297844\" class=\"unnumbered unstyled can-break\" summary=\"The product of two-thirds and the quantity 3 m minus 6 is equal to 5 minus m. Distribute. The result is 2 m minus 4 is equal to 5 minus m. Add m to both sides to get the variables only on the left. The result 2 m plus m minus 4 is equal to 5 minus m plus m. Simplify. The result is 3 m minus 4 is equal to 5. Add 4 to both sides to get constants only on the right. The result is 3 m minus 4 plus 4 is equal to 5 plus 4. Simplify. The result is 3 m is equal to 9. Divide both sides by 3. The result is 3 m divided by m is equal to 9 divided by 3. Simplify. The result is m is equal to 3. Check the solution in the original equation, the product of two-thirds and the quantity 3 m minus 6 is equal to 5 minus m. Let m be equal to 3. Is the product of two-thirds and the quantity 3 times 3 minus 6 equal to 5 minus 3? Is the product of two-thirds and the quantity 9 minus 6 equal to 2? Is the product of two-thirds and 3 equal to 2? 2 is equal 2. So, the solution checks.\"><tbody><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829743561\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836312094\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add <em data-effect=\"italics\">m<\/em> to both sides to get the variables only on the left.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836487266\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829833519\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add 4 to both sides to get constants only on the right.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836620710\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by three.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836597264\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836616426\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(m=3.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836341511\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833338727\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836511786\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836546981\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833024311\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833397229\"><div data-type=\"problem\" id=\"fs-id1167836359670\"><p id=\"fs-id1167836378444\">Solve: \\(\\frac{1}{3}\\left(6u+3\\right)=7-u.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833048494\"><p id=\"fs-id1167832977109\">\\(u=2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836549228\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836356384\"><div data-type=\"problem\" id=\"fs-id1167836514885\"><p id=\"fs-id1167836515025\">Solve: \\(\\frac{2}{3}\\left(9x-12\\right)=8+2x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836704262\"><p id=\"fs-id1167829742567\">\\(x=4\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836618895\">We can solve equations by getting all the <span data-type=\"term\" class=\"no-emphasis\">variable<\/span> terms to either side of the <span data-type=\"term\" class=\"no-emphasis\">equal sign<\/span>. By collecting the variable terms on the side where the <span data-type=\"term\" class=\"no-emphasis\">coefficient<\/span> of the variable is larger, we avoid working with some negatives. This will be a good strategy when we solve inequalities later in this chapter. It also helps us prevent errors with negatives.<\/p><div data-type=\"example\" id=\"fs-id1167836518499\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836551732\"><div data-type=\"problem\" id=\"fs-id1167829844120\"><p id=\"fs-id1167836610072\">Solve: \\(4\\left(x-1\\right)-2=5\\left(2x+3\\right)+6.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836392544\"><table id=\"fs-id1167829712046\" class=\"unnumbered unstyled can-break\" summary=\"The difference between 4 times the quantity x minus 1 and 2 is equal to the sum of 5 times the quantity 2 x plus 3 and 6. Distribute. The result is 4 x minus 4 minus 2 is equal to 19 x plus 15 plus 6. Combine like terms. The result is 4 x minus 6 is equal to 10 x plus 21. Subtract 4 x from each side to get the variables on the right side since 10 is less than 4. The result is 4 x minus 4 x minus 6 is equal to 10 x minus 4 x plus 21. Simplify. The result is negative 6 is equal to 6 x plus 21. Subtract 21 from each side to get the constants on the left. The result is negative 6 minus 21 is equal to 6 x plus 21 minus 21. Simplify. The result is negative 27 is equal to 6 x. Divide both sides by 6. Negative 27 divided by 6 is equal to 6 x divided by 6. Simplify. Negative nine-halves is equal to x. Check the solution in the original equation, the difference between 4 times the quantity x minus 1 and 2 is equal to the sum of 5 times the quantity 2 x plus 3 and 6 Let x be equal to negative nine-halves. Is the difference between 4 times the quantity negative nine-halves minus 1 and 2 equal to the sum of 5 times the quantity 2 times negative nine-halves plus 3 and 6? Is 4 times negative eleven-halves minus 2 equal to the sum of 5 times the quantity negative 9 plus 3 and 6? Is negative 22 minus 2 equal to 5 times negative 6 plus 6. Is negative 24 equal to negative 39 plus 6. Negative 24 is equal to negative 24. The solution checks.\"><tbody><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836628511\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836445035\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836788647\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract \\(4x\\) from each side to get the variables only on<div data-type=\"newline\"><br><\/div>the right since \\(10&gt;4.\\)<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836530991\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836683570\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract 21 from each side to get the constants on left.<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829651461\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829586684\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by 6.<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836513329\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836515198\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005o_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836319581\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><td><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(x=-\\frac{9}{2}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836415301\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><td><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836289713\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829644854\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><td><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836342493\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><td><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836501677\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836494104\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833020031\"><div data-type=\"problem\" id=\"fs-id1167836623059\"><p id=\"fs-id1167833350241\">Solve: \\(6\\left(p-3\\right)-7=5\\left(4p+3\\right)-12.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836608732\"><p id=\"fs-id1167836537468\">\\(p=-2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836662754\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836357330\"><div data-type=\"problem\" id=\"fs-id1167833379473\"><p id=\"fs-id1167836507118\">Solve: \\(8\\left(q+1\\right)-5=3\\left(2q-4\\right)-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836595652\"><p id=\"fs-id1167836284960\">\\(q=-8\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167836493446\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836706587\"><div data-type=\"problem\" id=\"fs-id1167836447977\"><p id=\"fs-id1167829596605\">Solve: \\(10\\left[3-8\\left(2s-5\\right)\\right]=15\\left(40-5s\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836312568\"><table id=\"fs-id1167836519944\" class=\"unnumbered unstyled can-break\" summary=\"10 times the difference of 3 and 8 times the quantity 2 s minus 5 is equal to 15 times the quantity 40 minus 5 s. Simplify from the innermost parentheses, 2 s minus 5, first. The result is 10 times the quantity 3 minus 16 s plus 40 is equal to 15 times the quantity 40 minus 5 s. Combine like terms in the brackets. The result is 10 times the quantity 43 minus 16 s is equal to 15 times the quantity 40 minus 5 s. Distribute. The result is 430 minus 160 s is equal to 600 minus 75 s. Add 160 to both sides to get the variables to the right. The result is 430 minus 160 s plus 160 s is equal to 600 minus 75 s plus 160 s. Simplify. The result is 430 is equal to 600 minus 85 s. Subtract 600 from both sides to get the constants to the left. The result is 430 minus 600 is equal to 600 plus 85 s minus 600. Simplify. The result is negative 170 is equal to 85 s. Divide. The result is negative 170 divided by 85 is equal to 85 s divided by 85. Simplify. The result is negative 2 is equal to s. Check the solution in the original equation, 10 times the difference of 3 and 8 times the quantity 2 s minus 5 is equal to 15 times the quantity 40 minus 5 s. Let s be equal to negative 2. Is 10 times the difference of 3 and 8 times the quantity 2 times negative 2 minus 5 equal to 15 times the quantity 40 minus 5 times negative 2? Is 10 times the difference of 3 and 8 times the quantity negative 4 minus 5 equal to 15 times the quantity 40 plus 10? Is 10 times the difference of 3 and 8 times negative 9 equal to 15 times 50? Is 10 times the quantity 3 plus 72 equal to 750? Is 10 times 75 equal to 750? 750 is equal to 750. The solution checks.\"><tbody><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981028\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify from the innermost parentheses first.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836287943\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Combine like terms in the brackets.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836391479\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836516709\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add \\(160s\\) to both sides to get the<div data-type=\"newline\"><br><\/div>variables to the right.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836512514\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836447826\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract 600 from both sides to get the<div data-type=\"newline\"><br><\/div>constants to the left.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829849401\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836323506\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006o_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by 85.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829749268\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006p_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829809882\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006q_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836522142\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(s=-2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767282\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836477510\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833346673\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836558664\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833369139\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829849430\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836717543\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836296969\"><div data-type=\"problem\" id=\"fs-id1167829696950\"><p id=\"fs-id1167836602664\">Solve: \\(6\\left[4-2\\left(7y-1\\right)\\right]=8\\left(13-8y\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833060180\"><p id=\"fs-id1167836527037\">\\(y=-\\frac{17}{5}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836732668\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829791737\"><div data-type=\"problem\" id=\"fs-id1167830121385\"><p id=\"fs-id1167832999497\">Solve: \\(12\\left[1-5\\left(4z-1\\right)\\right]=3\\left(24+11z\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836447538\"><p id=\"fs-id1167836673485\">\\(z=0\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833191628\"><h3 data-type=\"title\">Classify Equations<\/h3><p id=\"fs-id1167836575191\">Whether or not an equation is true depends on the value of the variable. The equation \\(7x+8=-13\\) is true when we replace the variable, <em data-effect=\"italics\">x<\/em>, with the value \\(-3,\\) but not true when we replace <em data-effect=\"italics\">x<\/em> with any other value. An equation like this is called a <span data-type=\"term\">conditional equation<\/span>. All the equations we have solved so far are conditional equations.<\/p><div data-type=\"note\" id=\"fs-id1167836579554\"><div data-type=\"title\">Conditional Equation<\/div><p id=\"fs-id1167836608511\">An equation that is true for one or more values of the variable and false for all other values of the variable is a <strong data-effect=\"bold\">conditional equation<\/strong>.<\/p><\/div><p id=\"fs-id1167829878831\">Now let\u2019s consider the equation \\(7y+14=7\\left(y+2\\right).\\) Do you recognize that the left side and the right side are equivalent? Let\u2019s see what happens when we solve for <em data-effect=\"italics\">y<\/em>.<\/p><p id=\"fs-id1167829908048\">Solve:<\/p><table id=\"fs-id1167836510870\" class=\"unnumbered unstyled\" summary=\"7 y plus 14 is equal to 7 times the quantity y plus 2. Distribute. The result is 7 y plus 14 is equal to 7 y plus 14. Subtract 7 y from each side to get the y\u2019s on one side. The result is 7 y minus 7 y plus 14 is equal to 7 y minus 7 y plus 14. When simplified, 14 is equal to 14. The y\u2019s are eliminated. But 14 is equal to 14 is true.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836690162\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833046908\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_007b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract \\(7y\\) to each side to get the \\(y\u2019\\text{s}\\) to one side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829702029\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_007c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify\u2014the <em data-effect=\"italics\">y<\/em>\u2019s are eliminated.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829747023\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_007d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">But \\(14=14\\) is true.<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836492919\">This means that the equation \\(7y+14=7\\left(y+2\\right)\\) is true for any value of <em data-effect=\"italics\">y<\/em>. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable is called an <span data-type=\"term\">identity<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167836665375\"><div data-type=\"title\">Identity<\/div><p id=\"fs-id1167830121972\">An equation that is true for any value of the variable is called an <strong data-effect=\"bold\">identity<\/strong>.<\/p><p id=\"fs-id1167824732887\">The solution of an identity is all real numbers.<\/p><\/div><p id=\"fs-id1167836684167\">What happens when we solve the equation \\(-8z=-8z+9?\\)<\/p><p id=\"fs-id1167829695652\">Solve:<\/p><table id=\"fs-id1167832994477\" class=\"unnumbered unstyled\" summary=\"Negative 8 z is equal to negative 8 z plus 9. Add 8 z to both sides to leave the constant alone on the right. The result is negative 8 z plus 8 z is equal to negative 8 z plus 8 z plus 9. When you simplify, the z\u2019s are eliminated. The result is 0 is equal to 9. But 0 is not equal to 9.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833369094\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add \\(8z\\) to both sides to leave the constant alone on the right.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836552465\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify\u2014the \\(z\u2019\\text{s}\\) are eliminated.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829650481\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_008c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">But \\(0\\ne 9.\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836666890\">Solving the equation \\(-8z=-8z+9\\) led to the false statement \\(0=9.\\) The equation \\(-8z=-8z+9\\) will not be true for any value of <em data-effect=\"italics\">z<\/em>. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a <span data-type=\"term\">contradiction<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167836326284\"><div data-type=\"title\">Contradiction<\/div><p id=\"fs-id1167833310062\">An equation that is false for all values of the variable is called a <strong data-effect=\"bold\">contradiction<\/strong>.<\/p><p id=\"fs-id1167829590351\">A contradiction has no solution.<\/p><\/div><p id=\"fs-id1167836481060\">The next few examples will ask us to classify an equation as conditional, an identity, or as a contradiction.<\/p><div data-type=\"example\" id=\"fs-id1167836666645\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836666647\"><div data-type=\"problem\" id=\"fs-id1167836666649\"><p id=\"fs-id1167836704660\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(6\\left(2n-1\\right)+3=2n-8+5\\left(2n+1\\right).\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167833059649\" class=\"unnumbered unstyled\" summary=\"Solve the equation, the product of 6 and the quantity 2 n minus 1 plus 3 is equal to 2 n minus 8 plus the product of 5 and the quantity 2 n plus 1. Distribute. The result is 12 n minus 6 plus 3 is equal to 2 n minus 8 plus 10 n plus 5. Combine like terms. The result is 12 n minus 3 is equal to 12 n minus 3. Subtract 12 n from each side to get the n\u2019s to one side. 12 n minus 12 n minus 3 is equal to 12 n minus 12 n minus 3. Simplify. Negative 3 is equal to negative 3. This is a true statement. The solution is an identity. The solution is all real numbers.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833057821\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829688803\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829720038\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract \\(12n\\) from each side to get the <em data-effect=\"italics\">n<\/em>\u2019s to one side.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829717176\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836390514\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">This is a true statement.<\/td><td data-valign=\"top\" data-align=\"left\">The equation is an identity.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The solution is all real numbers.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829853734\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836686437\"><div data-type=\"problem\" id=\"fs-id1167836686439\"><p id=\"fs-id1167836686442\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(4+9\\left(3x-7\\right)=-42x-13+23\\left(3x-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836544368\"><p id=\"fs-id1167836544370\">identity; all real numbers<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836512749\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836609322\"><div data-type=\"problem\" id=\"fs-id1167836609324\"><p id=\"fs-id1167836609327\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(8\\left(1-3x\\right)+15\\left(2x+7\\right)=2\\left(x+50\\right)+4\\left(x+3\\right)+1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750354\"><p id=\"fs-id1167836727987\">identity; all real numbers<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167836530045\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836530047\"><div data-type=\"problem\" id=\"fs-id1167836530049\"><p id=\"fs-id1167829905237\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(8+3\\left(a-4\\right)=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833339565\"><table id=\"fs-id1167829585697\" class=\"unnumbered unstyled\" summary=\"Solve the equation, 8 plus 3 times the quantity a minus 4 is equal to 0. Distribute. The result is 3 a minus 12 is equal to 0. Combine like terms. The result is 3 a minus 4 is equal to 0. Add 4 to both sides. The result is 3 a minus 4 plus 4 is equal to 0 plus 4. Simplify. The result is 3 a is equal to 4. Divide. The result is 3 a divided by 3 is equal to 4 divided by 3.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833224368\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829743910\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833060036\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add 4 to both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833349908\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836292392\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836697688\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836349627\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The equation is true when \\(a=\\frac{4}{3}.\\phantom{\\rule{2em}{0ex}}\\)<\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">This is a conditional equation.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">The solution is \\(a=\\frac{4}{3}.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836574042\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833059264\"><div data-type=\"problem\" id=\"fs-id1167833059266\"><p id=\"fs-id1167833059268\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(11\\left(q+3\\right)-5=19.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833326547\"><p id=\"fs-id1167836363673\">conditional equation; \\(q=-\\frac{9}{11}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836663177\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826170312\"><div data-type=\"problem\" id=\"fs-id1167826170314\"><p id=\"fs-id1167829598050\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(6+14\\left(k-8\\right)=95.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836699367\"><p id=\"fs-id1167836699370\">conditional equation; \\(k=\\frac{201}{14}\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167826211749\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167826211752\"><div data-type=\"problem\" id=\"fs-id1167826206384\"><p id=\"fs-id1167826206386\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(5m+3\\left(9+3m\\right)=2\\left(7m-11\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824755287\"><table id=\"fs-id1167836549906\" class=\"unnumbered unstyled\" summary=\"Solve the equation, 5 m plus 3 times the quantity 9 plus 3 m is equal to 2 times the quantity 7 m minus 11. Distribute. The result is 5 m plus 27 plus 9 m is equal to 14 m minus 22. Combine like terms. The result is 14 m plus 27 is equal to 14 m minus 22. Subtract 14 m from both sides. The result is 14 m plus 27 minus 14 m is equal to 14 m minus 22 minus 14 m. Simplify. The result is 27 is equal to negative 22. But 27 is not equal to negative 22. The equation is a contradiction. It has no solution.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836297006\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829692503\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833086337\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract \\(14m\\) from both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836791174\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836662933\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">But \\(27\\ne \\text{\u2212}22.\\)<\/td><td data-valign=\"top\" data-align=\"left\">The equation is a contradiction.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">It has no solution.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829850254\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836288569\"><div data-type=\"problem\" id=\"fs-id1167836288571\"><p id=\"fs-id1167836558040\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(12c+5\\left(5+3c\\right)=3\\left(9c-4\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836625095\"><p id=\"fs-id1167836625098\">contradiction; no solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829879479\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836506441\"><div data-type=\"problem\" id=\"fs-id1167836506444\"><p id=\"fs-id1167829621389\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \\(4\\left(7d+18\\right)=13\\left(3d-2\\right)-11d.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829785000\"><p id=\"fs-id1167829785002\">contradiction; no solution<\/p><\/div><\/div><\/div><p id=\"fs-id1167836579057\">We summarize the methods for classifying equations in the table.<\/p><table id=\"fs-id1167829686495\" summary=\"This table has three columns and four rows. The first row is a header row and it labels each column, \u201cType of equation,\u201d \u201cWhat happens when you solve it?\u201d and \u201cSolution.\u201d The second column is a header column and it labels each row \u201cConditional Equations,\u201d Identity,\u201d \u201cContradiction\u201d. In row two, the Conditional Equation is True for one or more values of the variables and false for all other values, and the Solution is One or more values. In row three, the Identity is True for any value of the variable, and the Solution is All real numbers. In row four, the Contradiction is False for all values of the variable, and the Solution is No Solution.\"><thead><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"center\">Type of equation<\/th><th data-valign=\"middle\" data-align=\"center\">What happens when you solve it?<\/th><th data-valign=\"middle\" data-align=\"center\">Solution<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Conditional Equation<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">True for one or more values of the variables and false for all other values<\/td><td data-valign=\"middle\" data-align=\"left\">One or more values<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Identity<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">True for any value of the variable<\/td><td data-valign=\"middle\" data-align=\"left\">All real numbers<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Contradiction<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">False for all values of the variable<\/td><td data-valign=\"middle\" data-align=\"left\">No solution<\/td><\/tr><\/tbody><\/table><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833047208\"><h3 data-type=\"title\">Solve Equations with Fraction or Decimal Coefficients<\/h3><p id=\"fs-id1167836289564\">We could use the General Strategy to solve the next example. This method would work fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.<\/p><p id=\"fs-id1167836543542\">We will apply the Multiplication Property of Equality and multiply both sides of an equation by the <span data-type=\"term\" class=\"no-emphasis\">least common denominator<\/span> (LCD) of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called <em data-effect=\"italics\">clearing<\/em> the equation of fractions.<\/p><p id=\"fs-id1167829791261\">To clear an equation of decimals, we think of all the decimals in their fraction form and then find the LCD of those denominators.<\/p><div data-type=\"example\" id=\"fs-id1167833239741\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve Equations with Fraction or Decimal Coefficients<\/div><div data-type=\"exercise\" id=\"fs-id1167833239743\"><div data-type=\"problem\" id=\"fs-id1167833239745\"><p id=\"fs-id1167829850197\">Solve: \\(\\frac{1}{12}x+\\frac{5}{6}=\\frac{3}{4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830121459\"><span data-type=\"media\" id=\"fs-id1167830121461\" data-alt=\"Step 1 is to find the least common denominator of all the fractions and decimals in the equation, one-twelfth x plus five-sixth is equal to three-fourths. What is the L C D of one-twelfth, five-sixths, and three-fourths? The L C D is equal to 12.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_012a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find the least common denominator of all the fractions and decimals in the equation, one-twelfth x plus five-sixth is equal to three-fourths. What is the L C D of one-twelfth, five-sixths, and three-fourths? The L C D is equal to 12.\"><\/span><span data-type=\"media\" id=\"fs-id1167836705888\" data-alt=\"Step 2 is multiply both sides of the equation by the L C D. This clears the fractions and decimals. Multiply both sides of the equation by the L C D, 12. The result is 12 times the quantity one-twelfth x plus five-sixths is equal to 12 times three-fourths. Use the Distributive Property. The result is 12 times one-twelfth x plus 12 times five-sixths is equal to 12 times three-fourths. Simplify. The result is x plus 10 is equal to 9. Notice there are no more fractions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_012b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is multiply both sides of the equation by the L C D. This clears the fractions and decimals. Multiply both sides of the equation by the L C D, 12. The result is 12 times the quantity one-twelfth x plus five-sixths is equal to 12 times three-fourths. Use the Distributive Property. The result is 12 times one-twelfth x plus 12 times five-sixths is equal to 12 times three-fourths. Simplify. The result is x plus 10 is equal to 9. Notice there are no more fractions.\"><\/span><span data-type=\"media\" id=\"fs-id1167836484473\" data-alt=\"Step 3 is to solve using the General Strategy for Solving Linear Equations. To isolate the variable term, subtract 10. The result is x plus 10 minus 10 is equal to 9 minus 10. Simplify. The result is x is equal to negative 1. Check the solution. Substitute negative into the original equation one-twelfth x plus five-sixths is equal to three-fourth. Is one-twelfth times negative 1 plus five-sixths equal to three-fourths? Is negative one-twelfth plus five-sixths equal to three-fourths? Is negative one-twelfth plus ten-twelfths equal to nine-twelfths? Is nine-twelfths equal to nine-twelfths? Yes. The solution checks.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_012c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve using the General Strategy for Solving Linear Equations. To isolate the variable term, subtract 10. The result is x plus 10 minus 10 is equal to 9 minus 10. Simplify. The result is x is equal to negative 1. Check the solution. Substitute negative into the original equation one-twelfth x plus five-sixths is equal to three-fourth. Is one-twelfth times negative 1 plus five-sixths equal to three-fourths? Is negative one-twelfth plus five-sixths equal to three-fourths? Is negative one-twelfth plus ten-twelfths equal to nine-twelfths? Is nine-twelfths equal to nine-twelfths? Yes. The solution checks.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824734928\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824734932\"><div data-type=\"problem\" id=\"fs-id1167824734934\"><p id=\"fs-id1167833350118\">Solve: \\(\\frac{1}{4}x+\\frac{1}{2}=\\frac{5}{8}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830121981\"><p id=\"fs-id1167836665098\">\\(x=\\frac{1}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167825824536\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167825824540\"><div data-type=\"problem\" id=\"fs-id1167836570287\"><p id=\"fs-id1167836570289\">Solve: \\(\\frac{1}{8}x+\\frac{1}{2}=\\frac{1}{4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824736161\"><p id=\"fs-id1167824736163\">\\(x=-2\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167824734048\">Notice in the previous example, once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve. We then used the <span data-type=\"term\" class=\"no-emphasis\">General Strategy for Solving Linear Equations<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167833380076\" class=\"howto\"><div data-type=\"title\">Solve Equations with Fraction or Decimal Coefficients.<\/div><ol id=\"fs-id1167824755205\" type=\"1\" class=\"stepwise\"><li>Find the least common denominator (LCD) of <em data-effect=\"italics\">all<\/em> the fractions and decimals (in fraction form) in the equation.<\/li><li>Multiply both sides of the equation by that LCD. This clears the fractions and decimals.<\/li><li>Solve using the General Strategy for Solving Linear Equations.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167833290811\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833290813\"><div data-type=\"problem\" id=\"fs-id1167826206284\"><p id=\"fs-id1167826206286\">Solve: \\(5=\\frac{1}{2}y+\\frac{2}{3}y-\\frac{3}{4}y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833021506\"><p id=\"fs-id1167833021508\">We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.<\/p><table id=\"fs-id1167833021512\" class=\"unnumbered unstyled\" summary=\"Find the L C D of all fractions in the equation, 5 is equal to one-half y plus two-thirds y minus three-fourths y. The L C D is 12. Multiply both sides of the equation by 12. The result is 12 times 5 is equal to 12 times the quantity (one-half y plus two-thirds y minus three-fourths y. Distribute. The result is 12 times 5 is equal to 12 times one-half y plus 12 times two-thirds y minus 12 times three-fourths y. Simplify and you\u2019ll notice there are no more fractions. The result is 60 is equal to 6 y plus 8 y minus 9 y. Combine like terms. The result is 60 is equal to 5 y. Divide by 5. The result is 60 divided by 5 is equal to 5 y divided by 5. Simplify. The result is 12 is equal to y. Check in the equation 5 is equal to one-half y plus two-thirds y minus three-fourths y. Let y be equal to 12. Is 5 equal to one-twelfth times 12 plus two-thirds times 12 minus three-fourths times 12? Is 5 equal to 6 plus 8 minus 9? 5 is equal to 5. The solution checks.\"><tbody><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Find the LCD of all fractions in the equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836552597\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">The LCD is 12.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply both sides of the equation by 12.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836532419\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829790054\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify\u2014notice, no more fractions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829752245\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836774782\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide by five.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836408301\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833056118\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824736432\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(y=12.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531546\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824732629\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829906141\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826131893\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826131897\"><div data-type=\"problem\" id=\"fs-id1167826131899\"><p id=\"fs-id1167826131902\">Solve: \\(7=\\frac{1}{2}x+\\frac{3}{4}x-\\frac{2}{3}x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836417312\"><p id=\"fs-id1167836417314\">\\(x=12\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836624962\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836444764\"><div data-type=\"problem\" id=\"fs-id1167836444766\"><p id=\"fs-id1167836444769\">Solve: \\(-1=\\frac{1}{2}u+\\frac{1}{4}u-\\frac{2}{3}u.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750370\"><p>\\(u=-12\\)<\/p><\/div><\/div><\/div><p>In the next example, we\u2019ll distribute before we clear the fractions.<\/p><div data-type=\"example\" id=\"fs-id1167829719791\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829719793\"><div data-type=\"problem\" id=\"fs-id1167829719795\"><p id=\"fs-id1167829809911\">Solve: \\(\\frac{1}{2}\\left(y-5\\right)=\\frac{1}{4}\\left(y-1\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829878369\"><table id=\"fs-id1167829878371\" class=\"unnumbered unstyled can-break\" summary=\"One-half times the quantity y minus 5 is equal to one-fourth times the quantity y minus 1. Distribute. The result is one-half times y minus one-half times 5 is equal to one-fourth times y minus one-fourth times 1. Simplify. The result is one-half times y minus five-halves is equal to one-fourth y minus one-fourth. Multiply by the L C D, 4. The result is 4 times the quantity one-half y minus five-halves is equal to 4 times the quantity one-fourth y minus one-fourth. Distribute. The result is 4 times one-half y minus 4 times five-halves is equal to 4 times one-fourth y minus 4 times one-fourth. Simplify. The result is 2 y minus 10 is equal to y minus 1. Collect the variables on the left by subtracting y from each side. The result is 2 y minus y minus 10 is equal to y minus y minus 1. Simplify. The result is y minus 10 is equal to negative 1. Collect the constants to the right by adding 10 to each side. The result is y minus 10 plus 10 is equal to negative 1 plus 10. Simplify. The result is y is equal 9. Multiply the L C D, 4, by each side of the equation one-half times the quantity y minus 5 is equal to one-fourth times the quantity y minus 1. The result is 4 times one-half times the quantity y minus 5 is equal to 4 times the one-fourth times the quantity y minus 1. Multiply 4 times the fractions. The result is 2 times the quantity y minus 5 is equal to 1 times the quantity y minus 1. Distribute. The result is 2 y minus 10 is equal to y minus 1. Collect the variables on the left by subtracting y from each side. The result is 2 y minus y minus 10 is equal to y minus y minus 1. Simplify. The result is y minus 10 is equal to negative 1. Collect the constants to the right by adding 10 to each side. The result is y minus 10 plus 10 is equal to negative 1 plus 10. Simplify. The result is y is equal to 9. Check the solution in the equation one-half times the quantity y minus 5 is equal to one-fourth times the quantity y minus 1. Let be equal to 9. Is one-half times the quantity 9 minus 5 equal to one-fourth times the quantity 9 minus 1. Finish the check on your own.\"><tbody><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836619773\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836556359\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836611989\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply by the LCD, four.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836610708\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833227116\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836519244\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the variables to the left.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829683931\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836487072\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the constants to the right.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836662868\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829748078\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"left\">An alternate way to solve this equation is to clear the fractions without distributing first. If you multiply the factors correctly, this method will be easier.<\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832994354\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply by the LCD, 4.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836521131\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply four times the fractions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767300\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014o_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833025436\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014p_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the variables to the left.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824736026\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014q_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767043\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014r_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the constants to the right.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829595324\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014s_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826132571\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014t_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826129441\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(y=9.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829893303\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Finish the check on your own.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824732997\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824733002\"><div data-type=\"problem\" id=\"fs-id1167824733004\"><p id=\"fs-id1167836575921\">Solve: \\(\\frac{1}{5}\\left(n+3\\right)=\\frac{1}{4}\\left(n+2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830075069\"><p id=\"fs-id1167830075071\">\\(n=2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829709190\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829709194\"><div data-type=\"problem\" id=\"fs-id1167829709196\"><p id=\"fs-id1167829709199\">Solve: \\(\\frac{1}{2}\\left(m-3\\right)=\\frac{1}{4}\\left(m-7\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836605824\"><p id=\"fs-id1167836605827\">\\(m=-1\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836605840\">When you multiply both sides of an equation by the LCD of the fractions, make sure you multiply each term by the LCD\u2014even if it does not contain a fraction.<\/p><div data-type=\"example\" id=\"fs-id1167836686849\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836686851\"><div data-type=\"problem\" id=\"fs-id1167836686853\"><p id=\"fs-id1167836686855\">Solve: \\(\\frac{4q+3}{2}+6=\\frac{3q+5}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824767436\"><table id=\"fs-id1167824767438\" class=\"unnumbered unstyled can-break\" summary=\"The quotient of the quantity 4 q plus 3 and 2 plus 6 is equal to the quotient of the quantity 3 q plus 5 and 4. Multiply both sides by the L C D, 4. The result is 4 times the quotient of the quantity 4 q plus 3 and 2 plus 6 is equal to 4 times the quotient of the quantity 3 q plus 5 and 4. Distribute. The result is 4 times the quotient of the quantity 4 q plus 3 and 2 plus 4 times 6 is equal to 4 times the quotient of the quantity 3 q plus 5 and 4. Simplify. The result is 2 times the quantity 4 q plus 3 plus 24 is equal to 3 q plus 5, which equals 8 q plus 6 plus 24 is equal to 3 q plus 5, which equals 8 q plus 30 is equal to 3 q plus 5. Collect the variables to the left by subtracting negative 3 q from each side. The result is 8 q minus 3 q plus 30 is equal to 3 q minus 3 q plus 5. Simplify. The result is 5 q plus 30 is equal to 5. Collect the constants to the right by subtracting 30 from each side. The result is 5 q plus 30 minus 30 is equal to 5 minus 30. Simplify. The result is 5 q is equal to negative 25. Divided both sides by 5. The result is 5 q divided by 5 is equal to negative 25 divided by 5. Simplify. The result is q is equal to negative 5. Check in the original equation, the quotient of the quantity 4 q plus 3 and 2 plus 6 is equal to the quotient of the quantity 3 q plus 5 and 4. Let q be equal to negative 5. Is the quotient of the quantity 4 times negative 5 plus 3 and 2 plus 6 is equal to the quotient of the quantity 3 times negative 5 plus 5 and 4? Finish the check on your own.\"><tbody><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829906110\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply both sides by the LCD, 4.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767056\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830121988\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767723\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767752\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767783\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the variables to the left.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830121483\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824736459\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the constants to the right.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826130023\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829783171\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by five.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829783202\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826132851\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824766815\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(q=-5.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830121691\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Finish the check on your own.<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833031476\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833031480\"><div data-type=\"problem\" id=\"fs-id1167833031482\"><p id=\"fs-id1167833031484\">Solve: \\(\\frac{3r+5}{6}+1=\\frac{4r+3}{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824736810\"><p id=\"fs-id1167824736813\">\\(r=1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824736826\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824736831\"><div data-type=\"problem\" id=\"fs-id1167824736833\"><p id=\"fs-id1167824736835\">Solve: \\(\\frac{2s+3}{2}+1=\\frac{3s+2}{4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832976524\"><p id=\"fs-id1167832976526\">\\(s=-8\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836575961\">Some equations have decimals in them. This kind of equation may occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, \\(0.7=\\frac{7}{10}\\) and \\(0.29=\\frac{29}{100}.\\) So, with an equation with decimals, we can use the same method we used to clear fractions\u2014multiply both sides of the equation by the <span data-type=\"term\" class=\"no-emphasis\">least common denominator<\/span>.<\/p><p id=\"fs-id1167836399279\">The next example uses an equation that is typical of the ones we will see in the money applications in a later section. Notice that we will clear all decimals by multiplying by the LCD of their fraction form.<\/p><div data-type=\"example\" id=\"fs-id1167836399284\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836399286\"><div data-type=\"problem\" id=\"fs-id1167836399289\"><p id=\"fs-id1167836399291\">Solve: \\(0.25x+0.05\\left(x+3\\right)=2.85.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830121736\"><p id=\"fs-id1167830121738\">Look at the decimals and think of the equivalent fractions:<\/p><div data-type=\"equation\" id=\"fs-id1167830961880\" class=\"unnumbered\" data-label=\"\">\\(0.25=\\frac{25}{100},\\phantom{\\rule{3em}{0ex}}0.05=\\frac{5}{100},\\phantom{\\rule{3em}{0ex}}2.85=2\\frac{85}{100}.\\)<\/div><p id=\"fs-id1167830121434\">Notice, the LCD is 100. By multiplying by the LCD we will clear the decimals from the equation.<\/p><table id=\"fs-id1167830121438\" class=\"unnumbered unstyled\" summary=\"0.25 x plus 0.05 times the quantity x plus 3 is equal to 2.85. Distribute. The result is 0.25 x plus 0.05 x plus 0.15 is equal to 2.85. Combine like terms. The result is 0.30 x plus 0.15 is equal to 2.85. To clear decimals, multiply by 100. The result is 100 times the quantity 0.30 x plus 0.15 is equal to 100 times 2.85. Distribute. The result is 30 x plus 15 is equal to 285. Subtract 15 from both sides. The result is 30 x plus 15 minus 15 is equal to 285 minus 15. Simplify. The result is 30 x is equal to 270. Divide each side by 30. The result is 30 x divided by 30 is equal to 270 is divided by 30. Simplify. The result is x is equal to 9.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829718925\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute first.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830074762\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830074789\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To clear decimals, multiply by 100.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836681112\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836681139\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract 15 from both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836536191\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832935674\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide by 30.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832935700\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829755725\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check it yourself by substituting \\(x=9\\) into the original equation.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829755768\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824767352\"><div data-type=\"problem\" id=\"fs-id1167824767355\"><p id=\"fs-id1167824767357\">Solve: \\(0.25n+0.05\\left(n+5\\right)=2.95.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824767393\"><p id=\"fs-id1167824767395\">\\(n=9\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824617115\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824617120\"><div data-type=\"problem\" id=\"fs-id1167824617122\"><p id=\"fs-id1167824617124\">Solve: \\(0.10d+0.05\\left(d-5\\right)=2.15.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824617160\"><p id=\"fs-id1167833086250\">\\(d=16\\)<\/p><\/div><\/div><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833086264\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167833086271\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">How to determine whether a number is a solution to an equation<\/strong><ol id=\"fs-id1167833086283\" type=\"1\" class=\"stepwise\"><li>Substitute the number in for the variable in the equation.<\/li><li>Simplify the expressions on both sides of the equation.<\/li><li>Determine whether the resulting equation is true.<div data-type=\"newline\"><br><\/div> If it is true, the number is a solution.<div data-type=\"newline\"><br><\/div> If it is not true, the number is not a solution.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to Solve Linear Equations Using a General Strategy<\/strong><ol id=\"fs-id1167829924631\" type=\"1\" class=\"stepwise\"><li>Simplify each side of the equation as much as possible.<div data-type=\"newline\"><br><\/div> Use the Distributive Property to remove any parentheses.<div data-type=\"newline\"><br><\/div> Combine like terms.<\/li><li>Collect all the variable terms on one side of the equation.<div data-type=\"newline\"><br><\/div> Use the Addition or Subtraction Property of Equality.<\/li><li>Collect all the constant terms on the other side of the equation.<div data-type=\"newline\"><br><\/div> Use the Addition or Subtraction Property of Equality.<\/li><li>Make the coefficient of the variable term equal to 1.<div data-type=\"newline\"><br><\/div> Use the Multiplication or Division Property of Equality.<div data-type=\"newline\"><br><\/div> State the solution to the equation.<\/li><li>Check the solution.<div data-type=\"newline\"><br><\/div> Substitute the solution into the original equation to make sure the result is a true statement.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to Solve Equations with Fraction or Decimal Coefficients<\/strong><ol id=\"fs-id1167824585347\" type=\"1\" class=\"stepwise\"><li>Find the least common denominator (LCD) of <em data-effect=\"italics\">all<\/em> the fractions and decimals (in fraction form) in the equation.<\/li><li>Multiply both sides of the equation by that LCD. This clears the fractions and decimals.<\/li><li>Solve using the General Strategy for Solving Linear Equations.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167824585373\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167824585377\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167824585384\"><strong data-effect=\"bold\">Solve Equations Using the General Strategy<\/strong><\/p><p id=\"fs-id1167824585391\">In the following exercises, determine whether the given values are solutions to the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167824585395\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832940265\"><p id=\"fs-id1167832940268\">\\(6y+10=12y\\)<\/p><p id=\"fs-id1167832940287\"><span class=\"token\">\u24d0<\/span>\\(y=\\frac{5}{3}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(y=-\\frac{1}{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833396489\"><p id=\"fs-id1167833396491\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833396505\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833396508\"><p id=\"fs-id1167833396510\">\\(4x+9=8x\\)<\/p><p id=\"fs-id1167833396529\"><span class=\"token\">\u24d0<\/span>\\(x=-\\frac{7}{8}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(x=\\frac{9}{4}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833274631\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833274633\"><p id=\"fs-id1167833274636\">\\(8u-1=6u\\)<\/p><p id=\"fs-id1167833274655\"><span class=\"token\">\u24d0<\/span>\\(u=-\\frac{1}{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(u=\\frac{1}{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836738172\"><p id=\"fs-id1167836738174\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836738188\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836738190\"><p id=\"fs-id1167836738193\">\\(9v-2=3v\\)<\/p><p id=\"fs-id1167836738212\"><span class=\"token\">\u24d0<\/span>\\(v=-\\frac{1}{3}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(v=\\frac{1}{3}\\)<\/div><\/div><p id=\"fs-id1167833050673\">In the following exercises, solve each linear equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167833050676\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833050678\"><p id=\"fs-id1167833050680\">\\(15\\left(y-9\\right)=-60\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833050705\"><p id=\"fs-id1167833050707\">\\(y=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833050720\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833050722\"><p id=\"fs-id1167836503222\">\\(-16\\left(3n+4\\right)=32\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836503264\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836503266\"><p id=\"fs-id1167836503268\">\\(\\text{\u2212}\\left(w-12\\right)=30\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829696611\"><p id=\"fs-id1167829696613\">\\(w=-18\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829696626\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829696628\"><p id=\"fs-id1167829696630\">\\(\\text{\u2212}\\left(t-19\\right)=28\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833041760\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833041762\"><p id=\"fs-id1167833041764\">\\(51+5\\left(4-q\\right)=56\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836619530\"><p id=\"fs-id1167836619532\">\\(q=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836619545\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836619547\"><p id=\"fs-id1167836619549\">\\(-6+6\\left(5-k\\right)=15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824734877\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167824734879\"><p id=\"fs-id1167824734881\">\\(3\\left(10-2x\\right)+54=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824734913\"><p id=\"fs-id1167833022376\">\\(x=14\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833022389\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833022391\"><p id=\"fs-id1167833022393\">\\(-2\\left(11-7x\\right)+54=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836663392\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836663395\"><p id=\"fs-id1167836663397\">\\(\\frac{2}{3}\\left(9c-3\\right)=22\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836663427\"><p id=\"fs-id1167836663429\">\\(c=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829712179\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829712181\"><p id=\"fs-id1167829712183\">\\(\\frac{3}{5}\\left(10x-5\\right)=27\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829593796\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829593799\"><p id=\"fs-id1167829593801\">\\(\\frac{1}{5}\\left(15c+10\\right)=c+7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829593837\"><p id=\"fs-id1167829593839\">\\(c=\\frac{5}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826129372\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826129374\"><p id=\"fs-id1167826129377\">\\(\\frac{1}{4}\\left(20d+12\\right)=d+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829712054\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829712056\"><p id=\"fs-id1167829712059\">\\(3\\left(4n-1\\right)-2=8n+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829712098\"><p id=\"fs-id1167829712100\">\\(n=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836712361\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836712363\"><p id=\"fs-id1167836712365\">\\(9\\left(2m-3\\right)-8=4m+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836697360\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836697362\"><p id=\"fs-id1167836697364\">\\(12+2\\left(5-3y\\right)=-9\\left(y-1\\right)-2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829685086\"><p id=\"fs-id1167829685088\">\\(y=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829685101\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829685103\"><p id=\"fs-id1167829685105\">\\(-15+4\\left(2-5y\\right)=-7\\left(y-4\\right)+4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833008516\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833008518\"><p id=\"fs-id1167833008521\">\\(5+6\\left(3s-5\\right)=-3+2\\left(8s-1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829712578\"><p id=\"fs-id1167829712580\">\\(s=10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836792617\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836792619\"><p id=\"fs-id1167836792621\">\\(-12+8\\left(x-5\\right)=-4+3\\left(5x-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833385456\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833385458\"><p id=\"fs-id1167833385460\">\\(4\\left(p-4\\right)-\\left(p+7\\right)=5\\left(p-3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836501600\"><p id=\"fs-id1167836501602\">\\(p=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836501615\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836501617\"><p id=\"fs-id1167836501619\">\\(3\\left(a-2\\right)-\\left(a+6\\right)=4\\left(a-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836611801\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836611803\"><p id=\"fs-id1167836611805\">\\(4\\left[5-8\\left(4c-3\\right)\\right]=12\\left(1-13c\\right)-8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824767510\"><p id=\"fs-id1167824616933\">\\(c=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824616946\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167824616948\"><p id=\"fs-id1167824616950\">\\(5\\left[9-2\\left(6d-1\\right)\\right]=11\\left(4-10d\\right)-139\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829828078\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829828080\"><p id=\"fs-id1167829828082\">\\(3\\left[-9+8\\left(4h-3\\right)\\right]=2\\left(5-12h\\right)-19\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830121956\"><p id=\"fs-id1167830121959\">\\(h=\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826132587\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826132589\"><p id=\"fs-id1167826132591\">\\(3\\left[-14+2\\left(15k-6\\right)\\right]=8\\left(3-5k\\right)-24\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836537423\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836537425\"><p id=\"fs-id1167836537427\">\\(5\\left[2\\left(m+4\\right)+8\\left(m-7\\right)\\right]=2\\left[3\\left(5+m\\right)-\\left(21-3m\\right)\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836363515\"><p id=\"fs-id1167836363517\">\\(m=6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836363530\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836363532\"><p id=\"fs-id1167836363534\">\\(10\\left[5\\left(n+1\\right)+4\\left(n-1\\right)\\right]=11\\left[7\\left(5+n\\right)-\\left(25-3n\\right)\\right]\\)<\/p><\/div><\/div><p id=\"fs-id1167829833141\"><strong data-effect=\"bold\">Classify Equations<\/strong><\/p><p id=\"fs-id1167829833146\">In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.<\/p><div data-type=\"exercise\" id=\"fs-id1167829833150\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829833153\"><p id=\"fs-id1167829833155\">\\(23z+19=3\\left(5z-9\\right)+8z+46\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833227023\"><p id=\"fs-id1167833227025\">identity; all real numbers<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833227030\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833227032\"><p id=\"fs-id1167833227034\">\\(15y+32=2\\left(10y-7\\right)-5y+46\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829878689\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829878691\"><p id=\"fs-id1167829878693\">\\(18\\left(5j-1\\right)+29=47\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833379578\"><p id=\"fs-id1167833379580\">conditional equation;\\(j=\\frac{2}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833379598\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833379600\"><p id=\"fs-id1167833379602\">\\(24\\left(3d-4\\right)+100=52\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829850955\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829850958\"><p id=\"fs-id1167829850960\">\\(22\\left(3m-4\\right)=8\\left(2m+9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833082173\"><p id=\"fs-id1167833082175\">conditional equation; \\(m=\\frac{16}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833082194\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833082196\"><p id=\"fs-id1167833082198\">\\(30\\left(2n-1\\right)=5\\left(10n+8\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829690644\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829690646\"><p id=\"fs-id1167829690648\">\\(7v+42=11\\left(3v+8\\right)-2\\left(13v-1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829791380\"><p id=\"fs-id1167829791382\">contradiction; no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829791388\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829791390\"><p id=\"fs-id1167829791392\">\\(18u-51=9\\left(4u+5\\right)-6\\left(3u-10\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833256292\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829743988\"><p id=\"fs-id1167829743990\">\\(45\\left(3y-2\\right)=9\\left(15y-6\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829744033\"><p id=\"fs-id1167829744035\">contradiction; no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744041\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829744043\"><p id=\"fs-id1167830122021\">\\(60\\left(2x-1\\right)=15\\left(8x+5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830122072\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830122074\"><p id=\"fs-id1167830122076\">\\(9\\left(14d+9\\right)+4d=13\\left(10d+6\\right)+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829930763\"><p id=\"fs-id1167829930765\">identity; all real numbers<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829789388\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829789390\"><p>\\(11\\left(8c+5\\right)-8c=2\\left(40c+25\\right)+5\\)<\/p><\/div><\/div><p id=\"fs-id1167833271817\"><strong data-effect=\"bold\">Solve Equations with Fraction or Decimal Coefficients<\/strong><\/p><p id=\"fs-id1167833271823\">In the following exercises, solve each equation with fraction coefficients.<\/p><div data-type=\"exercise\" id=\"fs-id1167833271826\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833271828\"><p id=\"fs-id1167833271830\">\\(\\frac{1}{4}x-\\frac{1}{2}=-\\frac{3}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833271860\"><p id=\"fs-id1167833271862\">\\(x=-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826025266\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826025268\"><p id=\"fs-id1167826025270\">\\(\\frac{3}{4}x-\\frac{1}{2}=\\frac{1}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830123905\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830123907\"><p id=\"fs-id1167830123909\">\\(\\frac{5}{6}y-\\frac{2}{3}=-\\frac{3}{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830123939\"><p id=\"fs-id1167830123941\">\\(y=-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830123954\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830123956\"><p id=\"fs-id1167830123958\">\\(\\frac{5}{6}y-\\frac{1}{3}=-\\frac{7}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829942608\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829942610\"><p id=\"fs-id1167829942612\">\\(\\frac{1}{2}a+\\frac{3}{8}=\\frac{3}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824578511\"><p id=\"fs-id1167824578513\">\\(a=\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824578529\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167824578531\"><p id=\"fs-id1167824578533\">\\(\\frac{5}{8}b+\\frac{1}{2}=-\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836737964\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836737966\"><p id=\"fs-id1167836737968\">\\(2=\\frac{1}{3}x-\\frac{1}{2}x+\\frac{2}{3}x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836705149\"><p id=\"fs-id1167836705151\">\\(x=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836705164\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836705166\"><p id=\"fs-id1167836705168\">\\(2=\\frac{3}{5}x-\\frac{1}{3}x+\\frac{2}{5}x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826205092\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826205094\"><p id=\"fs-id1167826205096\">\\(\\frac{1}{3}w+\\frac{5}{4}=w-\\frac{1}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826205128\"><p id=\"fs-id1167826205130\">\\(w=\\frac{9}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832951168\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832951170\"><p id=\"fs-id1167832951172\">\\(\\frac{1}{2}a-\\frac{1}{4}=\\frac{1}{6}a+\\frac{1}{12}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836440889\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836440891\"><p id=\"fs-id1167836440894\">\\(\\frac{1}{3}b+\\frac{1}{5}=\\frac{2}{5}b-\\frac{3}{5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836440931\"><p id=\"fs-id1167836440933\">\\(b=12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836690179\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836690181\"><p id=\"fs-id1167836690183\">\\(\\frac{1}{3}x+\\frac{2}{5}=\\frac{1}{5}x-\\frac{2}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836440167\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836440169\"><p id=\"fs-id1167836440171\">\\(\\frac{1}{4}\\left(p-7\\right)=\\frac{1}{3}\\left(p+5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836440215\"><p id=\"fs-id1167829717604\">\\(p=-41\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829717617\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829717619\"><p id=\"fs-id1167829717621\">\\(\\frac{1}{5}\\left(q+3\\right)=\\frac{1}{2}\\left(q-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833024688\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833024690\"><p id=\"fs-id1167833024692\">\\(\\frac{1}{2}\\left(x+4\\right)=\\frac{3}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833024723\"><p id=\"fs-id1167833024725\">\\(x=-\\frac{5}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836738014\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836738016\"><p id=\"fs-id1167836738018\">\\(\\frac{1}{3}\\left(x+5\\right)=\\frac{5}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829650961\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829650964\"><p id=\"fs-id1167829650966\">\\(\\frac{4n+8}{4}=\\frac{n}{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829650991\"><p id=\"fs-id1167829650993\">\\(n=-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829651006\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829651008\"><p id=\"fs-id1167836450454\">\\(\\frac{3p+6}{3}=\\frac{p}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836450495\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836450497\"><p id=\"fs-id1167836450499\">\\(\\frac{3x+4}{2}+1=\\frac{5x+10}{8}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836705276\"><p id=\"fs-id1167836705278\">\\(x=-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836705291\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836705293\"><p id=\"fs-id1167836705295\">\\(\\frac{10y-2}{3}+3=\\frac{10y+1}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836650234\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836650236\"><p id=\"fs-id1167836650238\">\\(\\frac{7u-1}{4}-1=\\frac{4u+8}{5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826131785\"><p id=\"fs-id1167826131787\">\\(u=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826131800\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826131802\"><p id=\"fs-id1167826131804\">\\(\\frac{3v-6}{2}+5=\\frac{11v-4}{5}\\)<\/p><\/div><\/div><p id=\"fs-id1167829850831\">In the following exercises, solve each equation with decimal coefficients.<\/p><div data-type=\"exercise\" id=\"fs-id1167829850834\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829850836\"><p id=\"fs-id1167829850838\">\\(0.4x+0.6=0.5x-1.2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836663977\"><p id=\"fs-id1167836663979\">\\(x=18\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836663992\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836663994\"><p id=\"fs-id1167836663996\">\\(0.7x+0.4=0.6x+2.4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836609439\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836609441\"><p id=\"fs-id1167836609443\">\\(0.9x-1.25=0.75x+1.75\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836609468\"><p id=\"fs-id1167836609470\">\\(x=20\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836609483\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836609485\"><p id=\"fs-id1167836609487\">\\(1.2x-0.91=0.8x+2.29\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833053741\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833053743\"><p id=\"fs-id1167833053746\">\\(0.05n+0.10\\left(n+8\\right)=2.15\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833129292\"><p id=\"fs-id1167833129294\">\\(n=9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833129307\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833129309\"><p id=\"fs-id1167833129311\">\\(0.05n+0.10\\left(n+7\\right)=3.55\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830074808\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830074810\"><p id=\"fs-id1167830074812\">\\(0.10d+0.25\\left(d+5\\right)=4.05\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830074843\"><p id=\"fs-id1167830074846\">\\(d=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830074858\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830074860\"><p id=\"fs-id1167830074863\">\\(0.10d+0.25\\left(d+7\\right)=5.25\\)<\/p><\/div><\/div><\/div><div class=\"everyday\" data-depth=\"2\" id=\"fs-id1167832935825\"><h4 data-type=\"title\">Everyday Math<\/h4><div data-type=\"exercise\" id=\"fs-id1167832935831\"><div data-type=\"problem\" id=\"fs-id1167832935833\"><p id=\"fs-id1167832935835\"><strong data-effect=\"bold\">Fencing<\/strong> Micah has 74 feet of fencing to make a dog run in his yard. He wants the length to be 2.5 feet more than the width. Find the length, <em data-effect=\"italics\">L<\/em>, by solving the equation \\(2L+2\\left(L-2.5\\right)=74.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836399644\"><p id=\"fs-id1167836399646\">\\(L=19.75\\) feet<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836399660\"><div data-type=\"problem\" id=\"fs-id1167836399662\"><p id=\"fs-id1167836399664\"><strong data-effect=\"bold\">Stamps<\/strong> Paula bought ?22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was eight less than the number of<\/p><div data-type=\"newline\"><br><\/div>49-cent stamps. Solve the equation<div data-type=\"newline\"><br><\/div>\\(0.49s+0.21\\text{\u200b}\\left(s-8\\right)\\text{\u200b}\\text{\u200b}=22.82\\) for <em data-effect=\"italics\">s<\/em>, to find the number of 49-cent stamps Paula bought.<\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836567810\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167836567818\"><div data-type=\"problem\" id=\"fs-id1167836567820\"><p id=\"fs-id1167836567822\">Using your own words, list the steps in the general strategy for solving linear equations.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836567827\"><p id=\"fs-id1167836567829\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836567834\"><div data-type=\"problem\" id=\"fs-id1167836567837\"><p id=\"fs-id1167836567839\">Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836567852\"><div data-type=\"problem\" id=\"fs-id1167836567854\"><p id=\"fs-id1167836567856\">What is the first step you take when solving the equation \\(3-7\\left(y-4\\right)=38?\\) Why is this your first step?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826172274\"><p id=\"fs-id1167826172277\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826172282\"><div data-type=\"problem\" id=\"fs-id1167826172284\"><p id=\"fs-id1167826172286\">If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826172299\"><div data-type=\"problem\" id=\"fs-id1167826172301\"><p id=\"fs-id1167833059092\">If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833059097\"><p id=\"fs-id1167833059099\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833059104\"><div data-type=\"problem\" id=\"fs-id1167833059106\"><p id=\"fs-id1167833059109\">For the equation \\(0.35x+2.1=3.85,\\) how do you clear the decimal?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833059140\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167833059145\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167836599405\" data-alt=\"This table has four columns and four rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve linear equations using a general strategy. In row 3, the I can was classify equations. In row 4, the I can was solve equations with fraction or decimal coefficients.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and four rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve linear equations using a general strategy. In row 3, the I can was classify equations. In row 4, the I can was solve equations with fraction or decimal coefficients.\"><\/span><p id=\"fs-id1167836599416\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p><p id=\"fs-id1167836599424\">\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p><p id=\"fs-id1167836599431\">\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p><p id=\"fs-id1167836599441\">\u2026no - I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167836599455\"><dt>conditional equation<\/dt><dd id=\"fs-id1167836599460\">An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.<\/dd><\/dl><dl id=\"fs-id1167833009488\"><dt>contradiction<\/dt><dd id=\"fs-id1167833009493\">An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.<\/dd><\/dl><dl id=\"fs-id1167833009498\"><dt>identity<\/dt><dd id=\"fs-id1167833009504\">An equation that is true for any value of the variable is called an Identity. The solution of an identity is all real numbers.<\/dd><\/dl><dl id=\"fs-id1167833009509\"><dt>linear equation<\/dt><dd id=\"fs-id1167833009515\">A linear equation is an equation in one variable that can be written, where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers and \\(a\\ne 0,\\) as \\(ax+b=0.\\)<\/dd><\/dl><dl id=\"fs-id1167829861282\"><dt>solution of an equation<\/dt><dd id=\"fs-id1167829861287\">A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Solve linear equations using a general strategy<\/li>\n<li>Classify equations<\/li>\n<li>Solve equations with fraction or decimal coefficients<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167825703510\" class=\"be-prepared\">\n<p id=\"fs-id1167833364840\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167833387029\" type=\"1\">\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15fc540e08424b70e443746e11275476_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#43;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829788421\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-633b91f5241bdf2e027f900469ebdd2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167836503089\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Find the LCD of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b589e486fc233b12d903a213a61a086_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-134d803fc1e11c50071777638b483ec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836518722\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836663302\">\n<h3 data-type=\"title\">Solve Linear Equations Using a General Strategy<\/h3>\n<p id=\"fs-id1167826205047\">Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that makes it a true statement. Any value of the variable that makes the equation true is called a <span data-type=\"term\">solution<\/span> to the equation. It is the answer to the puzzle!<\/p>\n<div data-type=\"note\" id=\"fs-id1167836341048\">\n<div data-type=\"title\">Solution of an Equation<\/div>\n<p id=\"fs-id1167833186674\">A <strong data-effect=\"bold\">solution<\/strong> of an equation is a value of a variable that makes a true statement when substituted into the equation.<\/p>\n<\/div>\n<p id=\"fs-id1167829634269\">To determine whether a number is a solution to an equation, we substitute the value for the variable in the equation. If the resulting equation is a true statement, then the number is a solution of the equation.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836487400\" class=\"howto\">\n<div data-type=\"title\">Determine Whether a Number is a Solution to an Equation.<\/div>\n<ol id=\"fs-id1167836305814\" type=\"1\" class=\"stepwise\">\n<li>Substitute the number for the variable in the equation.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Determine whether the resulting equation is true.\n<ul id=\"fs-id1167829788831\" data-bullet-style=\"bullet\">\n<li>If it is true, the number is a solution.<\/li>\n<li>If it is not true, the number is not a solution.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836388234\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836514463\">\n<div data-type=\"problem\" id=\"fs-id1167836623770\">\n<p id=\"fs-id1167833255992\">Determine whether the values are solutions to the equation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4874da994cbf11735089f7ae8f85eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#121;&#43;&#51;&#61;&#49;&#48;&#121;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836596040\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1afaea1734ef8e4d0ba11e34fb73696_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1b3d3ff6fe6c354b47d3da58935b3be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829851798\">\n<p id=\"fs-id1167836610243\">Since a solution to an equation is a value of the variable that makes the equation true, begin by substituting the value of the solution for the variable.<\/p>\n<p id=\"fs-id1167829720215\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829742512\" class=\"unnumbered unstyled\" summary=\"5 y plus 3 is equal to 10 y minus 4. Substitute three-fifths for y. Is the product of 5 and three-fifths plus 3 equal to the product of 10 and three-fifths minus 4. Multiply on each side of the equation. Is 3 plus 3 equal to 6 minus 4? Simplify on each side. 6 is not equal to 2. Since y is equal to three-fifths does not result in a true equation, y is equal to three-fifths is not a solution to the equation 5 y plus 3 is equal to 10 y minus 4.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836791181\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829811131\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836599202\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829691775\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981020\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_001e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836433419\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1afaea1734ef8e4d0ba11e34fb73696_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/> does not result in a true equation, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1afaea1734ef8e4d0ba11e34fb73696_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/> is not a solution to the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4874da994cbf11735089f7ae8f85eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#121;&#43;&#51;&#61;&#49;&#48;&#121;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167829714257\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836539230\" class=\"unnumbered unstyled\" summary=\"Substitute seven-fifths for y. Is the product of 5 and seven-fifths plus 3 equal to the product of 10 and seven-fifths minus 4. Multiply on each side of the equation. Is 7 plus 3 equal to 14 minus 4? Simplify on each side. 10 is equal to 10. Since y is equal to seven-fifths results in a true equation, y is equal to seven-fifths is a solution to the equation 5 y plus 3 is equal to 10 y minus 4.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836572094\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836320934\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829894394\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833058014\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829624689\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_002e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836352765\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1b3d3ff6fe6c354b47d3da58935b3be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/> results in a true equation, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1b3d3ff6fe6c354b47d3da58935b3be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/> is a solution to the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4874da994cbf11735089f7ae8f85eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#121;&#43;&#51;&#61;&#49;&#48;&#121;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836542370\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836492374\">\n<div data-type=\"problem\" id=\"fs-id1167836310213\">\n<p id=\"fs-id1167836621135\">Determine whether the values are solutions to the equation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9f464f8bbcb625f0e442a4622e5f2f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#121;&#43;&#50;&#61;&#54;&#121;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167833036727\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70b549b23d0269e0c4ccef7a5db57f1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d1b0ab367be28c3198c1ffa9c6a6c76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836692114\">\n<p id=\"fs-id1167826131759\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833008189\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836579562\">\n<div data-type=\"problem\" id=\"fs-id1167833339605\">\n<p id=\"fs-id1167833058165\">Determine whether the values are solutions to the equation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bca106cf0c3837b51e9c858882c00e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#50;&#61;&#50;&#120;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"127\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167829619789\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82aa8812cbd6ad536dcb3c4fddfef6d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd2171ab5e610de7079ae5a4090bee64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833347056\">\n<p id=\"fs-id1167836507257\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836335358\">There are many types of equations that we will learn to solve. In this section we will focus on a <span data-type=\"term\">linear equation<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836515647\">\n<div data-type=\"title\">Linear Equation<\/div>\n<p>A <strong data-effect=\"bold\">linear equation<\/strong> is an equation in one variable that can be written, where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d28cb478bd8b6abe7d9573551313d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> as:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829644900\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2c5a97201db17cfbef7b7550aa81e23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167833050177\">To solve a linear equation it is a good idea to have an overall strategy that can be used to solve any linear equation. In the next example, we will give the steps of a general strategy for solving any linear equation. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836432956\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Linear Equation Using a General Strategy<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836477553\">\n<div data-type=\"problem\" id=\"fs-id1167836333480\">\n<p id=\"fs-id1167836700210\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d52a466ca54488260c24f130392923f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#56;&#61;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"152\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836294565\"><span data-type=\"media\" id=\"fs-id1167833328021\" data-alt=\"Step 1 is to simplify each side of the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Use the Distributive Property. The equation first simplifies to 7 n minus 21 minus 8 is equal to negative 15. Then it simplifies to 7 n minus 29 is equal to negative 15. Notice that each side of the equation is now simplified as much as possible.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to simplify each side of the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Use the Distributive Property. The equation first simplifies to 7 n minus 21 minus 8 is equal to negative 15. Then it simplifies to 7 n minus 29 is equal to negative 15. Notice that each side of the equation is now simplified as much as possible.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836683590\" data-alt=\"Step 2 is to collect all variable terms on the left side of the equation, 7 n minus 29 is equal to negative 15. Notice there is nothing to do because all n\u2019s are on the left side.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to collect all variable terms on the left side of the equation, 7 n minus 29 is equal to negative 15. Notice there is nothing to do because all n\u2019s are on the left side.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829712528\" data-alt=\"Step 3 is to collect all constant terms on the other side of the equation, 7 n minus 29 is equal to negative 15. To get constants only on the right, add 29 to each side. The result is 7 n minus 29 plus 29 is equal to negative 15 plus 29. Simplify. The result is 7 n is equal to 14.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to collect all constant terms on the other side of the equation, 7 n minus 29 is equal to negative 15. To get constants only on the right, add 29 to each side. The result is 7 n minus 29 plus 29 is equal to negative 15 plus 29. Simplify. The result is 7 n is equal to 14.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829749807\" data-alt=\"Step 4 is to make the coefficient of the equation, 7 n is equal to 14, 1. Divide each side of the equation by 7. The result is 7 n divided by 7 is equal to 14 divided by 7. Simplify. The result is n is equal to 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to make the coefficient of the equation, 7 n is equal to 14, 1. Divide each side of the equation by 7. The result is 7 n divided by 7 is equal to 14 divided by 7. Simplify. The result is n is equal to 2.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836615156\" data-alt=\"Step 5 is to check the solution, n is equal to 2, by substituting into the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Is the product of 7 and the quantity 2 minus 3 minus 8 equal to negative 15? Subtract. Is 7 times negative 1 minus 8 equal to negative 15? Is negative 7 minus 8 equal to negative 15. Negative 15 is equal to negative 15. The solution checks.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_003e_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to check the solution, n is equal to 2, by substituting into the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Is the product of 7 and the quantity 2 minus 3 minus 8 equal to negative 15? Subtract. Is 7 times negative 1 minus 8 equal to negative 15? Is negative 7 minus 8 equal to negative 15. Negative 15 is equal to negative 15. The solution checks.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836389216\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836613031\">\n<div data-type=\"problem\" id=\"fs-id1167829694670\">\n<p id=\"fs-id1167836393244\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2eb14090b9305376cac0c05060639081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#61;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836628710\">\n<p id=\"fs-id1167836605160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ee3bb14bbe97a1114d697f8b45a9f94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829739303\">\n<div data-type=\"problem\" id=\"fs-id1167836599620\">\n<p id=\"fs-id1167836375852\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51f04925fe15bf894e8462cec1d3ae86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;&#61;&#45;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836512901\">\n<p id=\"fs-id1167836608138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-072bed0ebf929f9d7c14da365c8512a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829683897\">These steps are summarized in the <span data-type=\"term\" class=\"no-emphasis\">General Strategy for Solving Linear Equations<\/span> below.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836530302\" class=\"howto\">\n<div data-type=\"title\">Solve linear equations using a general strategy.<\/div>\n<ol id=\"fs-id1167836375366\" type=\"1\" class=\"stepwise\">\n<li>Simplify each side of the equation as much as possible.\n<div data-type=\"newline\"><\/div>\n<p> Use the Distributive Property to remove any parentheses.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Combine like terms.<\/li>\n<li>Collect all the variable terms on one side of the equation.\n<div data-type=\"newline\"><\/div>\n<p> Use the Addition or Subtraction Property of Equality.<\/li>\n<li>Collect all the constant terms on the other side of the equation.\n<div data-type=\"newline\"><\/div>\n<p> Use the Addition or Subtraction Property of Equality.<\/li>\n<li>Make the coefficient of the variable term equal to 1.\n<div data-type=\"newline\"><\/div>\n<p> Use the Multiplication or Division Property of Equality.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> State the solution to the equation.<\/li>\n<li>Check the solution.\n<div data-type=\"newline\"><\/div>\n<p> Substitute the solution into the original equation to make sure the result is a true statement.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836481206\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836389542\">\n<div data-type=\"problem\" id=\"fs-id1167829597819\">\n<p id=\"fs-id1167832930220\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d3b6912be61f3e0aace38e45bd2c5f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#109;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#45;&#109;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"155\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836424231\">\n<table id=\"fs-id1167836297844\" class=\"unnumbered unstyled can-break\" summary=\"The product of two-thirds and the quantity 3 m minus 6 is equal to 5 minus m. Distribute. The result is 2 m minus 4 is equal to 5 minus m. Add m to both sides to get the variables only on the left. The result 2 m plus m minus 4 is equal to 5 minus m plus m. Simplify. The result is 3 m minus 4 is equal to 5. Add 4 to both sides to get constants only on the right. The result is 3 m minus 4 plus 4 is equal to 5 plus 4. Simplify. The result is 3 m is equal to 9. Divide both sides by 3. The result is 3 m divided by m is equal to 9 divided by 3. Simplify. The result is m is equal to 3. Check the solution in the original equation, the product of two-thirds and the quantity 3 m minus 6 is equal to 5 minus m. Let m be equal to 3. Is the product of two-thirds and the quantity 3 times 3 minus 6 equal to 5 minus 3? Is the product of two-thirds and the quantity 9 minus 6 equal to 2? Is the product of two-thirds and 3 equal to 2? 2 is equal 2. So, the solution checks.\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829743561\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836312094\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add <em data-effect=\"italics\">m<\/em> to both sides to get the variables only on the left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836487266\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829833519\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add 4 to both sides to get constants only on the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836620710\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by three.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836597264\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836616426\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2631988c9ef41ca9652c02a83bef777b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836341511\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833338727\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836511786\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836546981\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_004e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833024311\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833397229\">\n<div data-type=\"problem\" id=\"fs-id1167836359670\">\n<p id=\"fs-id1167836378444\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0dc1ccb333d6452fe93ebd45d6d3d49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#117;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#45;&#117;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"143\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833048494\">\n<p id=\"fs-id1167832977109\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b267f9cd38a51f762c91bde8120afca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836549228\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836356384\">\n<div data-type=\"problem\" id=\"fs-id1167836514885\">\n<p id=\"fs-id1167836515025\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9e706d93e05fdc1f89d4b7fd3e360ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#120;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#43;&#50;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"161\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836704262\">\n<p id=\"fs-id1167829742567\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836618895\">We can solve equations by getting all the <span data-type=\"term\" class=\"no-emphasis\">variable<\/span> terms to either side of the <span data-type=\"term\" class=\"no-emphasis\">equal sign<\/span>. By collecting the variable terms on the side where the <span data-type=\"term\" class=\"no-emphasis\">coefficient<\/span> of the variable is larger, we avoid working with some negatives. This will be a good strategy when we solve inequalities later in this chapter. It also helps us prevent errors with negatives.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836518499\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836551732\">\n<div data-type=\"problem\" id=\"fs-id1167829844120\">\n<p id=\"fs-id1167836610072\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-149023e7803d0cb124ec5ae40e8e38dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#61;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"230\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836392544\">\n<table id=\"fs-id1167829712046\" class=\"unnumbered unstyled can-break\" summary=\"The difference between 4 times the quantity x minus 1 and 2 is equal to the sum of 5 times the quantity 2 x plus 3 and 6. Distribute. The result is 4 x minus 4 minus 2 is equal to 19 x plus 15 plus 6. Combine like terms. The result is 4 x minus 6 is equal to 10 x plus 21. Subtract 4 x from each side to get the variables on the right side since 10 is less than 4. The result is 4 x minus 4 x minus 6 is equal to 10 x minus 4 x plus 21. Simplify. The result is negative 6 is equal to 6 x plus 21. Subtract 21 from each side to get the constants on the left. The result is negative 6 minus 21 is equal to 6 x plus 21 minus 21. Simplify. The result is negative 27 is equal to 6 x. Divide both sides by 6. Negative 27 divided by 6 is equal to 6 x divided by 6. Simplify. Negative nine-halves is equal to x. Check the solution in the original equation, the difference between 4 times the quantity x minus 1 and 2 is equal to the sum of 5 times the quantity 2 x plus 3 and 6 Let x be equal to negative nine-halves. Is the difference between 4 times the quantity negative nine-halves minus 1 and 2 equal to the sum of 5 times the quantity 2 times negative nine-halves plus 3 and 6? Is 4 times negative eleven-halves minus 2 equal to the sum of 5 times the quantity negative 9 plus 3 and 6? Is negative 22 minus 2 equal to 5 times negative 6 plus 6. Is negative 24 equal to negative 39 plus 6. Negative 24 is equal to negative 24. The solution checks.\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836628511\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836445035\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836788647\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fa4d35b5c4377934245a54e07e43e7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: -1px;\" \/> from each side to get the variables only on<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the right since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10fd886781b0d98494d294f958b13e29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#62;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836530991\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836683570\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract 21 from each side to get the constants on left.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829651461\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829586684\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by 6.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836513329\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836515198\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005o_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836319581\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89c04c11242a4f402ca9f1e30ec3b357_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836415301\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836289713\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829644854\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836342493\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836501677\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_005f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836494104\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833020031\">\n<div data-type=\"problem\" id=\"fs-id1167836623059\">\n<p id=\"fs-id1167833350241\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61ffbb0fd381854e8a6b58f25783304e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#55;&#61;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"237\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836608732\">\n<p id=\"fs-id1167836537468\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e42b5341681aa874f384909b3cb6b14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836662754\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836357330\">\n<div data-type=\"problem\" id=\"fs-id1167833379473\">\n<p id=\"fs-id1167836507118\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-602b7c420f59b38e2f5f946e3fa82a75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#53;&#61;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#113;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"227\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836595652\">\n<p id=\"fs-id1167836284960\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c7be79160d0111891f868d7257bba89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836493446\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836706587\">\n<div data-type=\"problem\" id=\"fs-id1167836447977\">\n<p id=\"fs-id1167829596605\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad3663e1c06f3cf7a533bc6f0c91eb62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#91;&#51;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#115;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#49;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#48;&#45;&#53;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"255\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836312568\">\n<table id=\"fs-id1167836519944\" class=\"unnumbered unstyled can-break\" summary=\"10 times the difference of 3 and 8 times the quantity 2 s minus 5 is equal to 15 times the quantity 40 minus 5 s. Simplify from the innermost parentheses, 2 s minus 5, first. The result is 10 times the quantity 3 minus 16 s plus 40 is equal to 15 times the quantity 40 minus 5 s. Combine like terms in the brackets. The result is 10 times the quantity 43 minus 16 s is equal to 15 times the quantity 40 minus 5 s. Distribute. The result is 430 minus 160 s is equal to 600 minus 75 s. Add 160 to both sides to get the variables to the right. The result is 430 minus 160 s plus 160 s is equal to 600 minus 75 s plus 160 s. Simplify. The result is 430 is equal to 600 minus 85 s. Subtract 600 from both sides to get the constants to the left. The result is 430 minus 600 is equal to 600 plus 85 s minus 600. Simplify. The result is negative 170 is equal to 85 s. Divide. The result is negative 170 divided by 85 is equal to 85 s divided by 85. Simplify. The result is negative 2 is equal to s. Check the solution in the original equation, 10 times the difference of 3 and 8 times the quantity 2 s minus 5 is equal to 15 times the quantity 40 minus 5 s. Let s be equal to negative 2. Is 10 times the difference of 3 and 8 times the quantity 2 times negative 2 minus 5 equal to 15 times the quantity 40 minus 5 times negative 2? Is 10 times the difference of 3 and 8 times the quantity negative 4 minus 5 equal to 15 times the quantity 40 plus 10? Is 10 times the difference of 3 and 8 times negative 9 equal to 15 times 50? Is 10 times the quantity 3 plus 72 equal to 750? Is 10 times 75 equal to 750? 750 is equal to 750. The solution checks.\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981028\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify from the innermost parentheses first.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836287943\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Combine like terms in the brackets.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836391479\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836516709\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9218f8b1d5662b2542837bf393be575b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#48;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"34\" style=\"vertical-align: -1px;\" \/> to both sides to get the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>variables to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836512514\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836447826\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract 600 from both sides to get the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>constants to the left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829849401\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836323506\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006o_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by 85.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829749268\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006p_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829809882\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006q_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836522142\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab1cce951729ff0e7cd4d07f64f5fcae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"59\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767282\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836477510\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833346673\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836558664\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833369139\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829849430\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_006g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836717543\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836296969\">\n<div data-type=\"problem\" id=\"fs-id1167829696950\">\n<p id=\"fs-id1167836602664\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b7f497629f4ff22b5c6da8b6308c5b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#91;&#52;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#45;&#56;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"240\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833060180\">\n<p id=\"fs-id1167836527037\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16c59de61a143ecf13d5436658776dee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"63\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836732668\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829791737\">\n<div data-type=\"problem\" id=\"fs-id1167830121385\">\n<p id=\"fs-id1167832999497\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-063d2dbaf084fb0fd0ef8f8edc1fb4fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#91;&#49;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#122;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#52;&#43;&#49;&#49;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"256\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836447538\">\n<p id=\"fs-id1167836673485\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e381c273c9f7ca5b93029ee2e8bab16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833191628\">\n<h3 data-type=\"title\">Classify Equations<\/h3>\n<p id=\"fs-id1167836575191\">Whether or not an equation is true depends on the value of the variable. The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07d65de3bc010229f22614695d27af09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#120;&#43;&#56;&#61;&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"105\" style=\"vertical-align: -2px;\" \/> is true when we replace the variable, <em data-effect=\"italics\">x<\/em>, with the value <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f54336ab5f34f79c0d075fa22f60ec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> but not true when we replace <em data-effect=\"italics\">x<\/em> with any other value. An equation like this is called a <span data-type=\"term\">conditional equation<\/span>. All the equations we have solved so far are conditional equations.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836579554\">\n<div data-type=\"title\">Conditional Equation<\/div>\n<p id=\"fs-id1167836608511\">An equation that is true for one or more values of the variable and false for all other values of the variable is a <strong data-effect=\"bold\">conditional equation<\/strong>.<\/p>\n<\/div>\n<p id=\"fs-id1167829878831\">Now let\u2019s consider the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-322d4cf4ff46a280ec12dc94f71f24f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#121;&#43;&#49;&#52;&#61;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -4px;\" \/> Do you recognize that the left side and the right side are equivalent? Let\u2019s see what happens when we solve for <em data-effect=\"italics\">y<\/em>.<\/p>\n<p id=\"fs-id1167829908048\">Solve:<\/p>\n<table id=\"fs-id1167836510870\" class=\"unnumbered unstyled\" summary=\"7 y plus 14 is equal to 7 times the quantity y plus 2. Distribute. The result is 7 y plus 14 is equal to 7 y plus 14. Subtract 7 y from each side to get the y\u2019s on one side. The result is 7 y minus 7 y plus 14 is equal to 7 y minus 7 y plus 14. When simplified, 14 is equal to 14. The y\u2019s are eliminated. But 14 is equal to 14 is true.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836690162\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833046908\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_007b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-190eea1fe2fb9d508969d8a6808f6a1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\" \/> to each side to get the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a16408959b51a54ed25b4bfb340c64ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#39;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -4px;\" \/> to one side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829702029\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_007c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify\u2014the <em data-effect=\"italics\">y<\/em>\u2019s are eliminated.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829747023\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_007d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">But <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9709feb65b2597fd4ae1067562161194_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: -1px;\" \/> is true.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836492919\">This means that the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94e76dd43a92a4ca2a3597733f5e21db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#121;&#43;&#49;&#52;&#61;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -4px;\" \/> is true for any value of <em data-effect=\"italics\">y<\/em>. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable is called an <span data-type=\"term\">identity<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836665375\">\n<div data-type=\"title\">Identity<\/div>\n<p id=\"fs-id1167830121972\">An equation that is true for any value of the variable is called an <strong data-effect=\"bold\">identity<\/strong>.<\/p>\n<p id=\"fs-id1167824732887\">The solution of an identity is all real numbers.<\/p>\n<\/div>\n<p id=\"fs-id1167836684167\">What happens when we solve the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f5cd726ecc6b8f97af1ffa2c48a844a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#122;&#61;&#45;&#56;&#122;&#43;&#57;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"125\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167829695652\">Solve:<\/p>\n<table id=\"fs-id1167832994477\" class=\"unnumbered unstyled\" summary=\"Negative 8 z is equal to negative 8 z plus 9. Add 8 z to both sides to leave the constant alone on the right. The result is negative 8 z plus 8 z is equal to negative 8 z plus 8 z plus 9. When you simplify, the z\u2019s are eliminated. The result is 0 is equal to 9. But 0 is not equal to 9.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833369094\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ee57485ba4e430e6681b604c3d20ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> to both sides to leave the constant alone on the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836552465\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify\u2014the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89eaf2c65cc32f591c265e3915bc5bee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#39;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"21\" style=\"vertical-align: 0px;\" \/> are eliminated.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829650481\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_008c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">But <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d3c7d8e8616a72eb1011be1aa7a9509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#110;&#101;&#32;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836666890\">Solving the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83692d07b12baff2c489c4cf9af9cad0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#122;&#61;&#45;&#56;&#122;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"117\" style=\"vertical-align: -2px;\" \/> led to the false statement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6d0b8208ba4cfb1a48490c0dd82f050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: 0px;\" \/> The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83692d07b12baff2c489c4cf9af9cad0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#122;&#61;&#45;&#56;&#122;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"117\" style=\"vertical-align: -2px;\" \/> will not be true for any value of <em data-effect=\"italics\">z<\/em>. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a <span data-type=\"term\">contradiction<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836326284\">\n<div data-type=\"title\">Contradiction<\/div>\n<p id=\"fs-id1167833310062\">An equation that is false for all values of the variable is called a <strong data-effect=\"bold\">contradiction<\/strong>.<\/p>\n<p id=\"fs-id1167829590351\">A contradiction has no solution.<\/p>\n<\/div>\n<p id=\"fs-id1167836481060\">The next few examples will ask us to classify an equation as conditional, an identity, or as a contradiction.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836666645\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836666647\">\n<div data-type=\"problem\" id=\"fs-id1167836666649\">\n<p id=\"fs-id1167836704660\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15a2c29d1bd95a04a52c6a07a5194c8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#61;&#50;&#110;&#45;&#56;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"284\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167833059649\" class=\"unnumbered unstyled\" summary=\"Solve the equation, the product of 6 and the quantity 2 n minus 1 plus 3 is equal to 2 n minus 8 plus the product of 5 and the quantity 2 n plus 1. Distribute. The result is 12 n minus 6 plus 3 is equal to 2 n minus 8 plus 10 n plus 5. Combine like terms. The result is 12 n minus 3 is equal to 12 n minus 3. Subtract 12 n from each side to get the n\u2019s to one side. 12 n minus 12 n minus 3 is equal to 12 n minus 12 n minus 3. Simplify. Negative 3 is equal to negative 3. This is a true statement. The solution is an identity. The solution is all real numbers.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833057821\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829688803\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829720038\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2d0ae1007475ba0769cbc696734c8a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"28\" style=\"vertical-align: -1px;\" \/> from each side to get the <em data-effect=\"italics\">n<\/em>\u2019s to one side.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829717176\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836390514\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_009e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">This is a true statement.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The equation is an identity.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The solution is all real numbers.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829853734\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836686437\">\n<div data-type=\"problem\" id=\"fs-id1167836686439\">\n<p id=\"fs-id1167836686442\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67b4447304be38484fcf346d25b25faf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#43;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#50;&#120;&#45;&#49;&#51;&#43;&#50;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"323\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836544368\">\n<p id=\"fs-id1167836544370\">identity; all real numbers<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836512749\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836609322\">\n<div data-type=\"problem\" id=\"fs-id1167836609324\">\n<p id=\"fs-id1167836609327\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2607c1380d762c91fa24c7c42331acba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#51;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"402\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750354\">\n<p id=\"fs-id1167836727987\">identity; all real numbers<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836530045\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836530047\">\n<div data-type=\"problem\" id=\"fs-id1167836530049\">\n<p id=\"fs-id1167829905237\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c9a508a803b446e2e59b9e6abd731d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833339565\">\n<table id=\"fs-id1167829585697\" class=\"unnumbered unstyled\" summary=\"Solve the equation, 8 plus 3 times the quantity a minus 4 is equal to 0. Distribute. The result is 3 a minus 12 is equal to 0. Combine like terms. The result is 3 a minus 4 is equal to 0. Add 4 to both sides. The result is 3 a minus 4 plus 4 is equal to 0 plus 4. Simplify. The result is 3 a is equal to 4. Divide. The result is 3 a divided by 3 is equal to 4 divided by 3.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833224368\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829743910\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833060036\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add 4 to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833349908\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836292392\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836697688\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836349627\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_010g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The equation is true when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6212fb708485afcdfa484857a01a995_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/><\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">This is a conditional equation.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd5cad0ac28ddbd01f668561ce179e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836574042\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833059264\">\n<div data-type=\"problem\" id=\"fs-id1167833059266\">\n<p id=\"fs-id1167833059268\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f90141043c19367fa2de97ae6f72100_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#53;&#61;&#49;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833326547\">\n<p id=\"fs-id1167836363673\">conditional equation; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84bafe4f60cdea9e16f5ebe436b69b36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836663177\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826170312\">\n<div data-type=\"problem\" id=\"fs-id1167826170314\">\n<p id=\"fs-id1167829598050\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80531326255e8a0817881e755f637a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#43;&#49;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"151\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836699367\">\n<p id=\"fs-id1167836699370\">conditional equation; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf909d89d5ee4129e7addf10bf4e304f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#49;&#125;&#123;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"56\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167826211749\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167826211752\">\n<div data-type=\"problem\" id=\"fs-id1167826206384\">\n<p id=\"fs-id1167826206386\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8fc0d6df2343ab51bcd7109d95c1ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#109;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#43;&#51;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#109;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"247\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824755287\">\n<table id=\"fs-id1167836549906\" class=\"unnumbered unstyled\" summary=\"Solve the equation, 5 m plus 3 times the quantity 9 plus 3 m is equal to 2 times the quantity 7 m minus 11. Distribute. The result is 5 m plus 27 plus 9 m is equal to 14 m minus 22. Combine like terms. The result is 14 m plus 27 is equal to 14 m minus 22. Subtract 14 m from both sides. The result is 14 m plus 27 minus 14 m is equal to 14 m minus 22 minus 14 m. Simplify. The result is 27 is equal to negative 22. But 27 is not equal to negative 22. The equation is a contradiction. It has no solution.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836297006\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829692503\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833086337\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-928588ac288615757d2725f04a840305_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> from both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836791174\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836662933\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_011e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">But <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-424007639be4504a78fbcd8c7c297777_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#92;&#110;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">The equation is a contradiction.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">It has no solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829850254\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836288569\">\n<div data-type=\"problem\" id=\"fs-id1167836288571\">\n<p id=\"fs-id1167836558040\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08838070f6efae312bbff7fc6e32dd7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#99;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#51;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"222\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836625095\">\n<p id=\"fs-id1167836625098\">contradiction; no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829879479\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836506441\">\n<div data-type=\"problem\" id=\"fs-id1167836506444\">\n<p id=\"fs-id1167829621389\">Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7247899c6fe5d419ecc0ffdc1c552dbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#100;&#43;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#100;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#49;&#100;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"242\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829785000\">\n<p id=\"fs-id1167829785002\">contradiction; no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836579057\">We summarize the methods for classifying equations in the table.<\/p>\n<table id=\"fs-id1167829686495\" summary=\"This table has three columns and four rows. The first row is a header row and it labels each column, \u201cType of equation,\u201d \u201cWhat happens when you solve it?\u201d and \u201cSolution.\u201d The second column is a header column and it labels each row \u201cConditional Equations,\u201d Identity,\u201d \u201cContradiction\u201d. In row two, the Conditional Equation is True for one or more values of the variables and false for all other values, and the Solution is One or more values. In row three, the Identity is True for any value of the variable, and the Solution is All real numbers. In row four, the Contradiction is False for all values of the variable, and the Solution is No Solution.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"center\">Type of equation<\/th>\n<th data-valign=\"middle\" data-align=\"center\">What happens when you solve it?<\/th>\n<th data-valign=\"middle\" data-align=\"center\">Solution<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Conditional Equation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">True for one or more values of the variables and false for all other values<\/td>\n<td data-valign=\"middle\" data-align=\"left\">One or more values<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Identity<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">True for any value of the variable<\/td>\n<td data-valign=\"middle\" data-align=\"left\">All real numbers<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Contradiction<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">False for all values of the variable<\/td>\n<td data-valign=\"middle\" data-align=\"left\">No solution<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833047208\">\n<h3 data-type=\"title\">Solve Equations with Fraction or Decimal Coefficients<\/h3>\n<p id=\"fs-id1167836289564\">We could use the General Strategy to solve the next example. This method would work fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.<\/p>\n<p id=\"fs-id1167836543542\">We will apply the Multiplication Property of Equality and multiply both sides of an equation by the <span data-type=\"term\" class=\"no-emphasis\">least common denominator<\/span> (LCD) of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called <em data-effect=\"italics\">clearing<\/em> the equation of fractions.<\/p>\n<p id=\"fs-id1167829791261\">To clear an equation of decimals, we think of all the decimals in their fraction form and then find the LCD of those denominators.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833239741\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve Equations with Fraction or Decimal Coefficients<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833239743\">\n<div data-type=\"problem\" id=\"fs-id1167833239745\">\n<p id=\"fs-id1167829850197\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50dc5a1b4ee6a2d3019871bbe8cc3942_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"97\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830121459\"><span data-type=\"media\" id=\"fs-id1167830121461\" data-alt=\"Step 1 is to find the least common denominator of all the fractions and decimals in the equation, one-twelfth x plus five-sixth is equal to three-fourths. What is the L C D of one-twelfth, five-sixths, and three-fourths? The L C D is equal to 12.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_012a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find the least common denominator of all the fractions and decimals in the equation, one-twelfth x plus five-sixth is equal to three-fourths. What is the L C D of one-twelfth, five-sixths, and three-fourths? The L C D is equal to 12.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836705888\" data-alt=\"Step 2 is multiply both sides of the equation by the L C D. This clears the fractions and decimals. Multiply both sides of the equation by the L C D, 12. The result is 12 times the quantity one-twelfth x plus five-sixths is equal to 12 times three-fourths. Use the Distributive Property. The result is 12 times one-twelfth x plus 12 times five-sixths is equal to 12 times three-fourths. Simplify. The result is x plus 10 is equal to 9. Notice there are no more fractions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_012b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is multiply both sides of the equation by the L C D. This clears the fractions and decimals. Multiply both sides of the equation by the L C D, 12. The result is 12 times the quantity one-twelfth x plus five-sixths is equal to 12 times three-fourths. Use the Distributive Property. The result is 12 times one-twelfth x plus 12 times five-sixths is equal to 12 times three-fourths. Simplify. The result is x plus 10 is equal to 9. Notice there are no more fractions.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836484473\" data-alt=\"Step 3 is to solve using the General Strategy for Solving Linear Equations. To isolate the variable term, subtract 10. The result is x plus 10 minus 10 is equal to 9 minus 10. Simplify. The result is x is equal to negative 1. Check the solution. Substitute negative into the original equation one-twelfth x plus five-sixths is equal to three-fourth. Is one-twelfth times negative 1 plus five-sixths equal to three-fourths? Is negative one-twelfth plus five-sixths equal to three-fourths? Is negative one-twelfth plus ten-twelfths equal to nine-twelfths? Is nine-twelfths equal to nine-twelfths? Yes. The solution checks.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_012c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve using the General Strategy for Solving Linear Equations. To isolate the variable term, subtract 10. The result is x plus 10 minus 10 is equal to 9 minus 10. Simplify. The result is x is equal to negative 1. Check the solution. Substitute negative into the original equation one-twelfth x plus five-sixths is equal to three-fourth. Is one-twelfth times negative 1 plus five-sixths equal to three-fourths? Is negative one-twelfth plus five-sixths equal to three-fourths? Is negative one-twelfth plus ten-twelfths equal to nine-twelfths? Is nine-twelfths equal to nine-twelfths? Yes. The solution checks.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824734928\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824734932\">\n<div data-type=\"problem\" id=\"fs-id1167824734934\">\n<p id=\"fs-id1167833350118\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc56be7d8617403d8187d8c239dfa079_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830121981\">\n<p id=\"fs-id1167836665098\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46a3660db7e8d7648aa7837763fdd17d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167825824536\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167825824540\">\n<div data-type=\"problem\" id=\"fs-id1167836570287\">\n<p id=\"fs-id1167836570289\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a8ca8f1dde40c791c71be59ad59111f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824736161\">\n<p id=\"fs-id1167824736163\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167824734048\">Notice in the previous example, once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve. We then used the <span data-type=\"term\" class=\"no-emphasis\">General Strategy for Solving Linear Equations<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833380076\" class=\"howto\">\n<div data-type=\"title\">Solve Equations with Fraction or Decimal Coefficients.<\/div>\n<ol id=\"fs-id1167824755205\" type=\"1\" class=\"stepwise\">\n<li>Find the least common denominator (LCD) of <em data-effect=\"italics\">all<\/em> the fractions and decimals (in fraction form) in the equation.<\/li>\n<li>Multiply both sides of the equation by that LCD. This clears the fractions and decimals.<\/li>\n<li>Solve using the General Strategy for Solving Linear Equations.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167833290811\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833290813\">\n<div data-type=\"problem\" id=\"fs-id1167826206284\">\n<p id=\"fs-id1167826206286\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd299ba93d5950f9b3618eb24bb1dc6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#121;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"140\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833021506\">\n<p id=\"fs-id1167833021508\">We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.<\/p>\n<table id=\"fs-id1167833021512\" class=\"unnumbered unstyled\" summary=\"Find the L C D of all fractions in the equation, 5 is equal to one-half y plus two-thirds y minus three-fourths y. The L C D is 12. Multiply both sides of the equation by 12. The result is 12 times 5 is equal to 12 times the quantity (one-half y plus two-thirds y minus three-fourths y. Distribute. The result is 12 times 5 is equal to 12 times one-half y plus 12 times two-thirds y minus 12 times three-fourths y. Simplify and you\u2019ll notice there are no more fractions. The result is 60 is equal to 6 y plus 8 y minus 9 y. Combine like terms. The result is 60 is equal to 5 y. Divide by 5. The result is 60 divided by 5 is equal to 5 y divided by 5. Simplify. The result is 12 is equal to y. Check in the equation 5 is equal to one-half y plus two-thirds y minus three-fourths y. Let y be equal to 12. Is 5 equal to one-twelfth times 12 plus two-thirds times 12 minus three-fourths times 12? Is 5 equal to 6 plus 8 minus 9? 5 is equal to 5. The solution checks.\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Find the LCD of all fractions in the equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836552597\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">The LCD is 12.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply both sides of the equation by 12.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836532419\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829790054\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify\u2014notice, no more fractions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829752245\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836774782\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide by five.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836408301\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833056118\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824736432\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad3abb2747db31be8d42dfb663b8b668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531546\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824732629\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829906141\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_013d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826131893\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826131897\">\n<div data-type=\"problem\" id=\"fs-id1167826131899\">\n<p id=\"fs-id1167826131902\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75c82346bf577067011de4c25da45058_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"142\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836417312\">\n<p id=\"fs-id1167836417314\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d28a9fa084bc02bc048b780624acf003_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836624962\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836444764\">\n<div data-type=\"problem\" id=\"fs-id1167836444766\">\n<p id=\"fs-id1167836444769\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d533d9f86848ba3d58a2ee2f0ea009ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#117;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#117;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#117;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"155\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750370\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc54904014e9c4a846d5db75db22e2e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"65\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, we\u2019ll distribute before we clear the fractions.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829719791\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829719793\">\n<div data-type=\"problem\" id=\"fs-id1167829719795\">\n<p id=\"fs-id1167829809911\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22e82794dabbafcce88f2d9e9203a3da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"163\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829878369\">\n<table id=\"fs-id1167829878371\" class=\"unnumbered unstyled can-break\" summary=\"One-half times the quantity y minus 5 is equal to one-fourth times the quantity y minus 1. Distribute. The result is one-half times y minus one-half times 5 is equal to one-fourth times y minus one-fourth times 1. Simplify. The result is one-half times y minus five-halves is equal to one-fourth y minus one-fourth. Multiply by the L C D, 4. The result is 4 times the quantity one-half y minus five-halves is equal to 4 times the quantity one-fourth y minus one-fourth. Distribute. The result is 4 times one-half y minus 4 times five-halves is equal to 4 times one-fourth y minus 4 times one-fourth. Simplify. The result is 2 y minus 10 is equal to y minus 1. Collect the variables on the left by subtracting y from each side. The result is 2 y minus y minus 10 is equal to y minus y minus 1. Simplify. The result is y minus 10 is equal to negative 1. Collect the constants to the right by adding 10 to each side. The result is y minus 10 plus 10 is equal to negative 1 plus 10. Simplify. The result is y is equal 9. Multiply the L C D, 4, by each side of the equation one-half times the quantity y minus 5 is equal to one-fourth times the quantity y minus 1. The result is 4 times one-half times the quantity y minus 5 is equal to 4 times the one-fourth times the quantity y minus 1. Multiply 4 times the fractions. The result is 2 times the quantity y minus 5 is equal to 1 times the quantity y minus 1. Distribute. The result is 2 y minus 10 is equal to y minus 1. Collect the variables on the left by subtracting y from each side. The result is 2 y minus y minus 10 is equal to y minus y minus 1. Simplify. The result is y minus 10 is equal to negative 1. Collect the constants to the right by adding 10 to each side. The result is y minus 10 plus 10 is equal to negative 1 plus 10. Simplify. The result is y is equal to 9. Check the solution in the equation one-half times the quantity y minus 5 is equal to one-fourth times the quantity y minus 1. Let be equal to 9. Is one-half times the quantity 9 minus 5 equal to one-fourth times the quantity 9 minus 1. Finish the check on your own.\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836619773\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836556359\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836611989\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply by the LCD, four.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836610708\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833227116\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836519244\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the variables to the left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829683931\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836487072\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the constants to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836662868\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829748078\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\">An alternate way to solve this equation is to clear the fractions without distributing first. If you multiply the factors correctly, this method will be easier.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832994354\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply by the LCD, 4.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836521131\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply four times the fractions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767300\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014o_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833025436\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014p_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the variables to the left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824736026\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014q_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767043\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014r_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the constants to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829595324\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014s_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826132571\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014t_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826129441\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b9a38425cf3f48577a3ba0ec55f6482_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829893303\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_014b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Finish the check on your own.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824732997\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824733002\">\n<div data-type=\"problem\" id=\"fs-id1167824733004\">\n<p id=\"fs-id1167836575921\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fe23599cdd23e95628c37927736d800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"166\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830075069\">\n<p id=\"fs-id1167830075071\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b4c6cd9d27ba344abe355a47d378bb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829709190\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829709194\">\n<div data-type=\"problem\" id=\"fs-id1167829709196\">\n<p id=\"fs-id1167829709199\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4413f0448811658ff514b24c7d2b7a56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"176\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836605824\">\n<p id=\"fs-id1167836605827\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec3d084a8add5b00b06df1310c62281e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836605840\">When you multiply both sides of an equation by the LCD of the fractions, make sure you multiply each term by the LCD\u2014even if it does not contain a fraction.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836686849\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836686851\">\n<div data-type=\"problem\" id=\"fs-id1167836686853\">\n<p id=\"fs-id1167836686855\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ba86a48d45281148703262b516ceb8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#113;&#43;&#51;&#125;&#123;&#50;&#125;&#43;&#54;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#113;&#43;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824767436\">\n<table id=\"fs-id1167824767438\" class=\"unnumbered unstyled can-break\" summary=\"The quotient of the quantity 4 q plus 3 and 2 plus 6 is equal to the quotient of the quantity 3 q plus 5 and 4. Multiply both sides by the L C D, 4. The result is 4 times the quotient of the quantity 4 q plus 3 and 2 plus 6 is equal to 4 times the quotient of the quantity 3 q plus 5 and 4. Distribute. The result is 4 times the quotient of the quantity 4 q plus 3 and 2 plus 4 times 6 is equal to 4 times the quotient of the quantity 3 q plus 5 and 4. Simplify. The result is 2 times the quantity 4 q plus 3 plus 24 is equal to 3 q plus 5, which equals 8 q plus 6 plus 24 is equal to 3 q plus 5, which equals 8 q plus 30 is equal to 3 q plus 5. Collect the variables to the left by subtracting negative 3 q from each side. The result is 8 q minus 3 q plus 30 is equal to 3 q minus 3 q plus 5. Simplify. The result is 5 q plus 30 is equal to 5. Collect the constants to the right by subtracting 30 from each side. The result is 5 q plus 30 minus 30 is equal to 5 minus 30. Simplify. The result is 5 q is equal to negative 25. Divided both sides by 5. The result is 5 q divided by 5 is equal to negative 25 divided by 5. Simplify. The result is q is equal to negative 5. Check in the original equation, the quotient of the quantity 4 q plus 3 and 2 plus 6 is equal to the quotient of the quantity 3 q plus 5 and 4. Let q be equal to negative 5. Is the quotient of the quantity 4 times negative 5 plus 3 and 2 plus 6 is equal to the quotient of the quantity 3 times negative 5 plus 5 and 4? Finish the check on your own.\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829906110\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply both sides by the LCD, 4.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767056\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830121988\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767723\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767752\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824767783\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the variables to the left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830121483\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824736459\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Collect the constants to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826130023\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829783171\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by five.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829783202\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826132851\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824766815\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f724130c55d3b7cceda35c9f94390ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830121691\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_015b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Finish the check on your own.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833031476\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833031480\">\n<div data-type=\"problem\" id=\"fs-id1167833031482\">\n<p id=\"fs-id1167833031484\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54eadd6b9aa56c7a02e494e99943c64b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#114;&#43;&#53;&#125;&#123;&#54;&#125;&#43;&#49;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#114;&#43;&#51;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824736810\">\n<p id=\"fs-id1167824736813\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-020a271427b6fdedce143bb26dac3c27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824736826\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824736831\">\n<div data-type=\"problem\" id=\"fs-id1167824736833\">\n<p id=\"fs-id1167824736835\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e48337201ca6ce1be5f9a146f26a291c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#115;&#43;&#51;&#125;&#123;&#50;&#125;&#43;&#49;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#115;&#43;&#50;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832976524\">\n<p id=\"fs-id1167832976526\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5379eecbaed603d26952c65cb605cc35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836575961\">Some equations have decimals in them. This kind of equation may occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-098210eba5d5caed6c2a2e766e6a2618_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#55;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da8367b50ed9309b8f9bcdf57cc55132_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#57;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#57;&#125;&#123;&#49;&#48;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"84\" style=\"vertical-align: -7px;\" \/> So, with an equation with decimals, we can use the same method we used to clear fractions\u2014multiply both sides of the equation by the <span data-type=\"term\" class=\"no-emphasis\">least common denominator<\/span>.<\/p>\n<p id=\"fs-id1167836399279\">The next example uses an equation that is typical of the ones we will see in the money applications in a later section. Notice that we will clear all decimals by multiplying by the LCD of their fraction form.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836399284\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836399286\">\n<div data-type=\"problem\" id=\"fs-id1167836399289\">\n<p id=\"fs-id1167836399291\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc81ac1f0ea9c0f0d1627d3bf6dfe022_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#53;&#120;&#43;&#48;&#46;&#48;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#46;&#56;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"212\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830121736\">\n<p id=\"fs-id1167830121738\">Look at the decimals and think of the equivalent fractions:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167830961880\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c87e90180f23244f953711560fd5c0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#53;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#125;&#123;&#49;&#48;&#48;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#48;&#53;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#48;&#48;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#46;&#56;&#53;&#61;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#53;&#125;&#123;&#49;&#48;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"374\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"fs-id1167830121434\">Notice, the LCD is 100. By multiplying by the LCD we will clear the decimals from the equation.<\/p>\n<table id=\"fs-id1167830121438\" class=\"unnumbered unstyled\" summary=\"0.25 x plus 0.05 times the quantity x plus 3 is equal to 2.85. Distribute. The result is 0.25 x plus 0.05 x plus 0.15 is equal to 2.85. Combine like terms. The result is 0.30 x plus 0.15 is equal to 2.85. To clear decimals, multiply by 100. The result is 100 times the quantity 0.30 x plus 0.15 is equal to 100 times 2.85. Distribute. The result is 30 x plus 15 is equal to 285. Subtract 15 from both sides. The result is 30 x plus 15 minus 15 is equal to 285 minus 15. Simplify. The result is 30 x is equal to 270. Divide each side by 30. The result is 30 x divided by 30 is equal to 270 is divided by 30. Simplify. The result is x is equal to 9.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829718925\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute first.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830074762\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830074789\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To clear decimals, multiply by 100.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836681112\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836681139\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract 15 from both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836536191\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832935674\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide by 30.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832935700\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829755725\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_016i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check it yourself by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-300a345ef7b973d34879ac8e90555390_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> into the original equation.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829755768\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824767352\">\n<div data-type=\"problem\" id=\"fs-id1167824767355\">\n<p id=\"fs-id1167824767357\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bdf42bef07f10f8f5dc90adc292d8c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#53;&#110;&#43;&#48;&#46;&#48;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#46;&#57;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"213\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824767393\">\n<p id=\"fs-id1167824767395\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd7d88ae080606b45d7242624deeb556_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824617115\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824617120\">\n<div data-type=\"problem\" id=\"fs-id1167824617122\">\n<p id=\"fs-id1167824617124\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ede00fa13cba3c191ea43bc7e6313b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#49;&#48;&#100;&#43;&#48;&#46;&#48;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#46;&#49;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"210\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824617160\">\n<p id=\"fs-id1167833086250\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e104e29670d9ee9c267349e1d64ba59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833086264\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167833086271\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to determine whether a number is a solution to an equation<\/strong>\n<ol id=\"fs-id1167833086283\" type=\"1\" class=\"stepwise\">\n<li>Substitute the number in for the variable in the equation.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Determine whether the resulting equation is true.\n<div data-type=\"newline\"><\/div>\n<p> If it is true, the number is a solution.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> If it is not true, the number is not a solution.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to Solve Linear Equations Using a General Strategy<\/strong>\n<ol id=\"fs-id1167829924631\" type=\"1\" class=\"stepwise\">\n<li>Simplify each side of the equation as much as possible.\n<div data-type=\"newline\"><\/div>\n<p> Use the Distributive Property to remove any parentheses.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Combine like terms.<\/li>\n<li>Collect all the variable terms on one side of the equation.\n<div data-type=\"newline\"><\/div>\n<p> Use the Addition or Subtraction Property of Equality.<\/li>\n<li>Collect all the constant terms on the other side of the equation.\n<div data-type=\"newline\"><\/div>\n<p> Use the Addition or Subtraction Property of Equality.<\/li>\n<li>Make the coefficient of the variable term equal to 1.\n<div data-type=\"newline\"><\/div>\n<p> Use the Multiplication or Division Property of Equality.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> State the solution to the equation.<\/li>\n<li>Check the solution.\n<div data-type=\"newline\"><\/div>\n<p> Substitute the solution into the original equation to make sure the result is a true statement.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to Solve Equations with Fraction or Decimal Coefficients<\/strong>\n<ol id=\"fs-id1167824585347\" type=\"1\" class=\"stepwise\">\n<li>Find the least common denominator (LCD) of <em data-effect=\"italics\">all<\/em> the fractions and decimals (in fraction form) in the equation.<\/li>\n<li>Multiply both sides of the equation by that LCD. This clears the fractions and decimals.<\/li>\n<li>Solve using the General Strategy for Solving Linear Equations.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167824585373\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167824585377\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167824585384\"><strong data-effect=\"bold\">Solve Equations Using the General Strategy<\/strong><\/p>\n<p id=\"fs-id1167824585391\">In the following exercises, determine whether the given values are solutions to the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167824585395\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832940265\">\n<p id=\"fs-id1167832940268\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-847611efeb1c8585ecdf8250b6ee4e1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#121;&#43;&#49;&#48;&#61;&#49;&#50;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167832940287\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c73c4e368071ff3adc8be5a15b82d1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d73f92d367c6005c529d143de2bb9ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833396489\">\n<p id=\"fs-id1167833396491\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833396505\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833396508\">\n<p id=\"fs-id1167833396510\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d441cf407ea3167d0aab04427495848_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#57;&#61;&#56;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"92\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167833396529\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6554a1fcb192bf6b7d717df683abcaf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a746d40e74b8b485839830f2947b03cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833274631\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833274633\">\n<p id=\"fs-id1167833274636\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13e31ec00b50f03c08edcc4c18946159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#117;&#45;&#49;&#61;&#54;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"92\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1167833274655\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06fa5bef24cb67d936d55c3027f6ed42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4f2f342eda648164f05380c4f4e9909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836738172\">\n<p id=\"fs-id1167836738174\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836738188\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836738190\">\n<p id=\"fs-id1167836738193\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8305c74b8fcc345b4d84fa781c5fa364_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#118;&#45;&#50;&#61;&#51;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"90\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167836738212\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d5a78b569a6e33a5191ad7747ba2e6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b14a9dcc4984ef10068c6e640940bb4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167833050673\">In the following exercises, solve each linear equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833050676\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833050678\">\n<p id=\"fs-id1167833050680\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c22aedb016138c667792538df410de77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833050705\">\n<p id=\"fs-id1167833050707\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39af43abd99adaf051fde7775af522c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833050720\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833050722\">\n<p id=\"fs-id1167836503222\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f3c72f350ab467ffe0f0fb6f898055a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#110;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836503264\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836503266\">\n<p id=\"fs-id1167836503268\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb9f1e79c1a3aaa7e9fe119dc38ac1b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829696611\">\n<p id=\"fs-id1167829696613\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32a1e3b3fc67b1822e7abdb495807a81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"69\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829696626\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829696628\">\n<p id=\"fs-id1167829696630\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fade9a5cb67941599d26b88fc4840af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833041760\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833041762\">\n<p id=\"fs-id1167833041764\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ba862be8d34b7ca9935146678fdd169_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#49;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836619530\">\n<p id=\"fs-id1167836619532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a668f463751568b57520c44997425e59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836619545\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836619547\">\n<p id=\"fs-id1167836619549\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4a1e176ee3df57cdc716d537d95536d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824734877\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824734879\">\n<p id=\"fs-id1167824734881\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c861dcd54ab34068791c7c3b773fa6fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#45;&#50;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824734913\">\n<p id=\"fs-id1167833022376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d25a6066c3d9b7c9fbb343dc35223881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833022389\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833022391\">\n<p id=\"fs-id1167833022393\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61555c4b64641935ae4e5f36d35794c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#45;&#55;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;&#52;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"169\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836663392\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836663395\">\n<p id=\"fs-id1167836663397\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c8875338505aeab253db3a2d6ba9f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"113\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836663427\">\n<p id=\"fs-id1167836663429\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bf86f02f34fdc43ccbd3c178c88cd29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829712179\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829712181\">\n<p id=\"fs-id1167829712183\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c6778206856deacd41e6ca52d0e2465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829593796\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829593799\">\n<p id=\"fs-id1167829593801\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c8e2b875c08b0f604900832b9aa4677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#99;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#99;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"152\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829593837\">\n<p id=\"fs-id1167829593839\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c947adffdf3d24c907d6cfea4cec3634_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826129372\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826129374\">\n<p id=\"fs-id1167826129377\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18cd53e7012a97df14a39456731429e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#48;&#100;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#100;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"155\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829712054\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829712056\">\n<p id=\"fs-id1167829712059\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e30d6516f5be48a6691035690823b22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#61;&#56;&#110;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"180\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829712098\">\n<p id=\"fs-id1167829712100\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b4c6cd9d27ba344abe355a47d378bb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836712361\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836712363\">\n<p id=\"fs-id1167836712365\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1123bf80d5a2074196f796b034eb9cfa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#109;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#56;&#61;&#52;&#109;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"190\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836697360\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836697362\">\n<p id=\"fs-id1167836697364\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e6066d99323a0c3c78568a598c1c53a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#51;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"245\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829685086\">\n<p id=\"fs-id1167829685088\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee7bf59b42611ad4534a6d8ca47648e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829685101\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829685103\">\n<p id=\"fs-id1167829685105\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7ce9fe4340808af06519d8546bf2887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"260\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833008516\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833008518\">\n<p id=\"fs-id1167833008521\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bc95ed554c5cfafa75df5168fca296f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#115;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#115;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"244\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829712578\">\n<p id=\"fs-id1167829712580\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca4b34faf3628df3ec9928dfaec81f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836792617\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836792619\">\n<p id=\"fs-id1167836792621\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff63c98a7f9aa005146dea765cae380e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#43;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"261\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833385456\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833385458\">\n<p id=\"fs-id1167833385460\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3b15aaa1a162c3279a4d1abb9f7ac84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"228\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836501600\">\n<p id=\"fs-id1167836501602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d256f1215455e8def588a5c0351d037b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836501615\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836501617\">\n<p id=\"fs-id1167836501619\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd8fae98485fbb1f4f0f77270f2de131_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"229\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836611801\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836611803\">\n<p id=\"fs-id1167836611805\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3695d9300d573587b8b870479aa67d86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#91;&#53;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#99;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#49;&#51;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"270\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824767510\">\n<p id=\"fs-id1167824616933\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc2df6832c4b35231873132e3ca5233a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824616946\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824616948\">\n<p id=\"fs-id1167824616950\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e1972621ba4d504930077a2071eb404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#91;&#57;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#100;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#49;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#49;&#48;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#51;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"291\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829828078\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829828080\">\n<p id=\"fs-id1167829828082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a836f079ce18886f3c5b2b9ec5d1414_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#57;&#43;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#104;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#49;&#50;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"289\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830121956\">\n<p id=\"fs-id1167830121959\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7a00009c3dd83e7d2b2eb3dccfbf155_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826132587\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826132589\">\n<p id=\"fs-id1167826132591\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0fa6c2210c0a3e9182933d9c0739ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#49;&#52;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#107;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#53;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"297\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836537423\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836537425\">\n<p id=\"fs-id1167836537427\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97c74a5cfa81a0b5b93c5bcb1acdb2dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#91;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#91;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#49;&#45;&#51;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"401\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836363515\">\n<p id=\"fs-id1167836363517\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7fc72e96811a37d7cae11c613f3a11e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836363530\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836363532\">\n<p id=\"fs-id1167836363534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e966e6e2166cc773b87e3b94653e232_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#91;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#49;&#49;&#92;&#108;&#101;&#102;&#116;&#91;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#53;&#45;&#51;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"398\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829833141\"><strong data-effect=\"bold\">Classify Equations<\/strong><\/p>\n<p id=\"fs-id1167829833146\">In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829833150\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829833153\">\n<p id=\"fs-id1167829833155\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7ce986b0d63bf172b5b50580fec8407_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#51;&#122;&#43;&#49;&#57;&#61;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#122;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#56;&#122;&#43;&#52;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"243\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833227023\">\n<p id=\"fs-id1167833227025\">identity; all real numbers<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833227030\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833227032\">\n<p id=\"fs-id1167833227034\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f55eb067af949a802acd60f35360bfea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#121;&#43;&#51;&#50;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#53;&#121;&#43;&#52;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"252\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829878689\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829878691\">\n<p id=\"fs-id1167829878693\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65dde71c01b897de0cb87bbf9e9024a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#106;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#57;&#61;&#52;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"162\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833379578\">\n<p id=\"fs-id1167833379580\">conditional equation;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2fc297426c694ebad6d95fd256ff0989_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833379598\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833379600\">\n<p id=\"fs-id1167833379602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91cb635beafbcda898641fa6e58f42a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#100;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#48;&#48;&#61;&#53;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829850955\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829850958\">\n<p id=\"fs-id1167829850960\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa725fce9133997e83090992f143bf77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#109;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#109;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"193\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833082173\">\n<p id=\"fs-id1167833082175\">conditional equation; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-946adbd84ae22a1225f75ab3b763215a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833082194\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833082196\">\n<p id=\"fs-id1167833082198\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78f624cfb8729e1f6dbaf77dac4c1a44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#110;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"192\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829690644\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829690646\">\n<p id=\"fs-id1167829690648\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8aa6ebb7c53c764d603259e8d98c3b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#118;&#43;&#52;&#50;&#61;&#49;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#118;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#118;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"268\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829791380\">\n<p id=\"fs-id1167829791382\">contradiction; no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829791388\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829791390\">\n<p id=\"fs-id1167829791392\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7a9d738756cacde18dde84b1210834b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#117;&#45;&#53;&#49;&#61;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#117;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#117;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"270\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833256292\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829743988\">\n<p id=\"fs-id1167829743990\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96347630ab10fd00fd1bb0eb4cd1361d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829744033\">\n<p id=\"fs-id1167829744035\">contradiction; no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829744041\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829744043\">\n<p id=\"fs-id1167830122021\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f55040c6298e6dd12168ea6905d59009_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"191\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830122072\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830122074\">\n<p id=\"fs-id1167830122076\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37bfd769504360307fc45e958a82be15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#100;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#100;&#61;&#49;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#100;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"269\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829930763\">\n<p id=\"fs-id1167829930765\">identity; all real numbers<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829789388\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829789390\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a7032457b280e8ce83839fd35af1bd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#99;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#56;&#99;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#48;&#99;&#43;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"262\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833271817\"><strong data-effect=\"bold\">Solve Equations with Fraction or Decimal Coefficients<\/strong><\/p>\n<p id=\"fs-id1167833271823\">In the following exercises, solve each equation with fraction coefficients.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833271826\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833271828\">\n<p id=\"fs-id1167833271830\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a185e60f2cd972529b99c4b0c1a47e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833271860\">\n<p id=\"fs-id1167833271862\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826025266\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826025268\">\n<p id=\"fs-id1167826025270\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ed483b76905875e85e5fe6463fabcf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830123905\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830123907\">\n<p id=\"fs-id1167830123909\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbd20501e49d02e6d210bb5bd41b35da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#121;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830123939\">\n<p id=\"fs-id1167830123941\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f371b4e77462e28d9f6119571c92982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830123954\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830123956\">\n<p id=\"fs-id1167830123958\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c30fba2b3e425310b3fd68caeebd51f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#121;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829942608\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829942610\">\n<p id=\"fs-id1167829942612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e962812f729a7ee96ae344b8493fa18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#97;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"83\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824578511\">\n<p id=\"fs-id1167824578513\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9d2e3abea9a3f3ae5ab7b620ec653c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824578529\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824578531\">\n<p id=\"fs-id1167824578533\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59eb4d28115c406948d58eb17d2fc8d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#98;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836737964\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836737966\">\n<p id=\"fs-id1167836737968\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2303c1ecc22d5122bf2e1f317fd7457e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"138\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836705149\">\n<p id=\"fs-id1167836705151\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836705164\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836705166\">\n<p id=\"fs-id1167836705168\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b3e7bd56c45a86e8bbe0939802e4319_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"138\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826205092\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826205094\">\n<p id=\"fs-id1167826205096\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af58f6a456ac83557e944256ae5e6339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#119;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#61;&#119;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"122\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826205128\">\n<p id=\"fs-id1167826205130\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51d9906e6bc19d8bb463b525233b4a7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832951168\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832951170\">\n<p id=\"fs-id1167832951172\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1df42adb38246b08a54f13980cabdb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#97;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#97;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"132\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836440889\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836440891\">\n<p id=\"fs-id1167836440894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de5aa589096bf06f6fb30ad665eae087_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#98;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#98;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836440931\">\n<p id=\"fs-id1167836440933\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-694df2df3e556212b9e18844a3f5336e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"48\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836690179\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836690181\">\n<p id=\"fs-id1167836690183\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de742f9a7e50b5e84cac462893187a58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836440167\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836440169\">\n<p id=\"fs-id1167836440171\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3fcabe95fed981b55855f17301e42b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"155\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836440215\">\n<p id=\"fs-id1167829717604\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f61d1641ff52f9b3fa0c1ee120875ae9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#52;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829717617\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829717619\">\n<p id=\"fs-id1167829717621\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62fc10e92b5ba7e95ba0c4173bc11cf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"154\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833024688\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833024690\">\n<p id=\"fs-id1167833024692\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf129d8eac2fcb3c302f27b77a6895ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833024723\">\n<p id=\"fs-id1167833024725\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4bbe1878177bfeeef0086a1141568be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836738014\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836738016\">\n<p id=\"fs-id1167836738018\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f44e3fec8393a23a88132a3e4b686b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829650961\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829650964\">\n<p id=\"fs-id1167829650966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c2d9a908685633cf02913059b4cec02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#110;&#43;&#56;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829650991\">\n<p id=\"fs-id1167829650993\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac12605a682d0ded7ba7f4ac4a0ab159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829651006\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829651008\">\n<p id=\"fs-id1167836450454\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f113841e9b6f269da70198c048716da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#112;&#43;&#54;&#125;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836450495\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836450497\">\n<p id=\"fs-id1167836450499\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ec76dfc02aa1eec51e87720ee0b807c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#43;&#52;&#125;&#123;&#50;&#125;&#43;&#49;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#43;&#49;&#48;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836705276\">\n<p id=\"fs-id1167836705278\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836705291\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836705293\">\n<p id=\"fs-id1167836705295\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75d9ebc4546c18e1965d915df2fda1b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#121;&#45;&#50;&#125;&#123;&#51;&#125;&#43;&#51;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#121;&#43;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"136\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836650234\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836650236\">\n<p id=\"fs-id1167836650238\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2a916d775e4ad6992f7c9a198ab06a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#117;&#45;&#49;&#125;&#123;&#52;&#125;&#45;&#49;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#117;&#43;&#56;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"123\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826131785\">\n<p id=\"fs-id1167826131787\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c9362b4d3f1e72137a095026128b158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826131800\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826131802\">\n<p id=\"fs-id1167826131804\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ac862a24d974fc520f72c85629a2404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#118;&#45;&#54;&#125;&#123;&#50;&#125;&#43;&#53;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#118;&#45;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"129\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829850831\">In the following exercises, solve each equation with decimal coefficients.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829850834\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829850836\">\n<p id=\"fs-id1167829850838\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a66cd2298c95ab63dcdcec86eeffb4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#52;&#120;&#43;&#48;&#46;&#54;&#61;&#48;&#46;&#53;&#120;&#45;&#49;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"177\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836663977\">\n<p id=\"fs-id1167836663979\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92c6693f5b085b97d573203dd74cba05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836663992\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836663994\">\n<p id=\"fs-id1167836663996\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d88a8ab7ad9ebb20b39dfffa46662625_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#55;&#120;&#43;&#48;&#46;&#52;&#61;&#48;&#46;&#54;&#120;&#43;&#50;&#46;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"178\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836609439\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836609441\">\n<p id=\"fs-id1167836609443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47e75602511d99351865331e3c3996a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#57;&#120;&#45;&#49;&#46;&#50;&#53;&#61;&#48;&#46;&#55;&#53;&#120;&#43;&#49;&#46;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"204\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836609468\">\n<p id=\"fs-id1167836609470\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-679269b04081838b5b220c088fa962fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836609483\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836609485\">\n<p id=\"fs-id1167836609487\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2eba473e4ce60e4e8ebd4b376e0271d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#50;&#120;&#45;&#48;&#46;&#57;&#49;&#61;&#48;&#46;&#56;&#120;&#43;&#50;&#46;&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"195\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833053741\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833053743\">\n<p id=\"fs-id1167833053746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-810f5d059cc83f7f539c94d640861efb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#53;&#110;&#43;&#48;&#46;&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#46;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"208\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833129292\">\n<p id=\"fs-id1167833129294\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd7d88ae080606b45d7242624deeb556_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833129307\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833129309\">\n<p id=\"fs-id1167833129311\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-830593ccc0fef21cc2a8ae490fc0fe2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#53;&#110;&#43;&#48;&#46;&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#46;&#53;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"208\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830074808\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830074810\">\n<p id=\"fs-id1167830074812\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a5fafd19795dcbf31d03f586035d884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#49;&#48;&#100;&#43;&#48;&#46;&#50;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#46;&#48;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"205\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830074843\">\n<p id=\"fs-id1167830074846\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bf30ff1853c57417244c77f5bac3f31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830074858\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830074860\">\n<p id=\"fs-id1167830074863\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28b3aa6a0e5cbe797df54a0572488234_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#49;&#48;&#100;&#43;&#48;&#46;&#50;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"205\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"everyday\" data-depth=\"2\" id=\"fs-id1167832935825\">\n<h4 data-type=\"title\">Everyday Math<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167832935831\">\n<div data-type=\"problem\" id=\"fs-id1167832935833\">\n<p id=\"fs-id1167832935835\"><strong data-effect=\"bold\">Fencing<\/strong> Micah has 74 feet of fencing to make a dog run in his yard. He wants the length to be 2.5 feet more than the width. Find the length, <em data-effect=\"italics\">L<\/em>, by solving the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07c160308a3de8a36f36837959d40d55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#76;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#76;&#45;&#50;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"170\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836399644\">\n<p id=\"fs-id1167836399646\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6e030d7724b1a1210b8d6c10e2ad7b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#49;&#57;&#46;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"76\" style=\"vertical-align: -1px;\" \/> feet<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836399660\">\n<div data-type=\"problem\" id=\"fs-id1167836399662\">\n<p id=\"fs-id1167836399664\"><strong data-effect=\"bold\">Stamps<\/strong> Paula bought ?22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was eight less than the number of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>49-cent stamps. Solve the equation<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52ed42d8d84480dbb380d0f28ef2931a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#52;&#57;&#115;&#43;&#48;&#46;&#50;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#115;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#61;&#50;&#50;&#46;&#56;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"215\" style=\"vertical-align: -4px;\" \/> for <em data-effect=\"italics\">s<\/em>, to find the number of 49-cent stamps Paula bought.<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836567810\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167836567818\">\n<div data-type=\"problem\" id=\"fs-id1167836567820\">\n<p id=\"fs-id1167836567822\">Using your own words, list the steps in the general strategy for solving linear equations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836567827\">\n<p id=\"fs-id1167836567829\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836567834\">\n<div data-type=\"problem\" id=\"fs-id1167836567837\">\n<p id=\"fs-id1167836567839\">Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836567852\">\n<div data-type=\"problem\" id=\"fs-id1167836567854\">\n<p id=\"fs-id1167836567856\">What is the first step you take when solving the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f9627b8cb39b52d4b9dd60bc6453670_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#56;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -4px;\" \/> Why is this your first step?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826172274\">\n<p id=\"fs-id1167826172277\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826172282\">\n<div data-type=\"problem\" id=\"fs-id1167826172284\">\n<p id=\"fs-id1167826172286\">If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826172299\">\n<div data-type=\"problem\" id=\"fs-id1167826172301\">\n<p id=\"fs-id1167833059092\">If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833059097\">\n<p id=\"fs-id1167833059099\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833059104\">\n<div data-type=\"problem\" id=\"fs-id1167833059106\">\n<p id=\"fs-id1167833059109\">For the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cbbdde1dc4dc9b97c25f0bbc3f10de4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#51;&#53;&#120;&#43;&#50;&#46;&#49;&#61;&#51;&#46;&#56;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -4px;\" \/> how do you clear the decimal?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833059140\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167833059145\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836599405\" data-alt=\"This table has four columns and four rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve linear equations using a general strategy. In row 3, the I can was classify equations. In row 4, the I can was solve equations with fraction or decimal coefficients.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_01_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and four rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve linear equations using a general strategy. In row 3, the I can was classify equations. In row 4, the I can was solve equations with fraction or decimal coefficients.\" \/><\/span><\/p>\n<p id=\"fs-id1167836599416\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p>\n<p id=\"fs-id1167836599424\">\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p>\n<p id=\"fs-id1167836599431\">\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p>\n<p id=\"fs-id1167836599441\">\u2026no &#8211; I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167836599455\">\n<dt>conditional equation<\/dt>\n<dd id=\"fs-id1167836599460\">An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833009488\">\n<dt>contradiction<\/dt>\n<dd id=\"fs-id1167833009493\">An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833009498\">\n<dt>identity<\/dt>\n<dd id=\"fs-id1167833009504\">An equation that is true for any value of the variable is called an Identity. The solution of an identity is all real numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833009509\">\n<dt>linear equation<\/dt>\n<dd id=\"fs-id1167833009515\">A linear equation is an equation in one variable that can be written, where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d28cb478bd8b6abe7d9573551313d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c0ddda3a39cc88982724f83d6a78b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"85\" style=\"vertical-align: -2px;\" \/><\/dd>\n<\/dl>\n<dl id=\"fs-id1167829861282\">\n<dt>solution of an equation<\/dt>\n<dd id=\"fs-id1167829861287\">A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1136","chapter","type-chapter","status-publish","hentry"],"part":991,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1136","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1136\/revisions"}],"predecessor-version":[{"id":15146,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1136\/revisions\/15146"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/991"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1136\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=1136"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=1136"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=1136"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=1136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}