3.3 Solve Equations with Variables and Constants on Both Sides
Learning Objectives
By the end of this section, you will be able to:
- Solve an equation with constants on both sides
- Solve an equation with variables on both sides
- Solve an equation with variables and constants on both sides
- Solve equations using a general strategy
Solve an Equation with Constants on Both Sides
You may have noticed that in all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we’ll see how to solve equations where the variable terms and/or constant terms are on both sides of the equation.
Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to get all the variable terms together on one side of the equation and the constant terms together on the other side.
By doing this, we will transform the equation that started with variables and constants on both sides into the form . We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.
EXAMPLE 1
Solve: .
Solution
In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side. We’ll write the labels above the equation to help us remember what goes where.
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| Since the left side is the variable side, the 6 is out of place. We must “undo” adding 6 by subtracting 6, and to keep the equality we must subtract 6 from both sides. Use the Subtraction Property of Equality. | |
| Simplify. | |
| Now all the s are on the left and the constant on the right. | |
| Use the Division Property of Equality. | |
| Simplify. | |
| Check: | |
| Let . | |
TRY IT 1.1
Solve: .
Show answer
x = −4
TRY IT 1.2
Solve: .
Show answer
a = −8
EXAMPLE 2
Solve: .
Solution
Notice that the variable is only on the left side of the equation, so this will be the variable side and the right side will be the constant side. Since the left side is the variable side, the is out of place. It is subtracted from the , so to ‘undo’ subtraction, add to both sides.
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| Add 7 to both sides. | |
| Simplify. | |
| The variables are now on one side and the constants on the other. | |
| Divide both sides by 2. | |
| Simplify. | |
| Check: | |
| Substitute: . | |
TRY IT 2.1
Solve: .
Show answer
y = 5
TRY IT 2.2
Solve: .
Show answer
m = 9
Solve an Equation with Variables on Both Sides
What if there are variables on both sides of the equation? We will start like we did above—choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side. Remember, what you do to the left side of the equation, you must do to the right side too.
EXAMPLE 3
Solve: .
Here the variable, , is on both sides, but the constants appear only on the right side, so let’s make the right side the “constant” side. Then the left side will be the “variable” side.
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| We don’t want any variables on the right, so subtract the . | |
| Simplify. | |
| We have all the variables on one side and the constants on the other. We have solved the equation. | |
| Check: | |
| Substitute 7 for . | |
TRY IT 3.1
Solve: .
Show answer
n = 10
TRY IT 3.2
Solve: .
Show answer
c = 1
EXAMPLE 4
Solve: .
Solution
The only constant, , is on the left side of the equation and variable, , is on both sides. Let’s leave the constant on the left and collect the variables to the right.
| Subtract from both sides. | |
| Simplify. | |
| We have the variables on the right and the constants on the left. Divide both sides by 2. | |
| Simplify. | |
| Rewrite with the variable on the left. | |
| Check: Let . |
TRY IT 4.1
Solve: .
Show answer
p = −7
TRY IT 4.2
Solve: .
Show answer
m = −3
EXAMPLE 5
Solve: .
Solution
The only constant, , is on the right, so let the left side be the variable side.
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| Remove the from the right side by adding to both sides. | |
| Simplify. | |
| All the variables are on the left and the constants are on the right. Divide both sides by 8. | |
| Simplify. | |
| Check: Substitute . |
TRY IT 5.1
Solve: .
Show answer
j = 2
TRY IT 5.2
Solve: .
Show answer
h = 1
Solve Equations with Variables and Constants on Both Sides
The next example will be the first to have variables and constants on both sides of the equation. As we did before, we’ll collect the variable terms to one side and the constants to the other side.
EXAMPLE 6
Solve: .
Solution
Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are and . Since is greater than , make the left side the variable side and so the right side will be the constant side.
| Collect the variable terms to the left side by subtracting from both sides. | |
| Simplify. | |
| Now, collect the constants to the right side by subtracting 5 from both sides. | |
| Simplify. | |
| The solution is . | |
| Check: Let . |
TRY IT 6.1
Solve: .
Show answer
x = −1
TRY IT 6.2
Solve: .
Show answer
y = 4
We’ll summarize the steps we took so you can easily refer to them.
HOW TO: Solve an Equation with Variables and Constants on Both Sides
- Choose one side to be the variable side and then the other will be the constant side.
- Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
- Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
- Make the coefficient of the variable , using the Multiplication or Division Property of Equality.
- Check the solution by substituting it into the original equation.
It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.
EXAMPLE 7
Solve: .
Solution
We have on the left and on the right. Since > , make the left side the “variable” side.
| We don’t want variables on the right side—add to both sides to leave only constants on the right. | |
| Combine like terms. | |
| We don’t want any constants on the left side, so add 2 to both sides. | |
| Simplify. | |
| The variable term is on the left and the constant term is on the right. To get the coefficient of to be one, divide both sides by 9. |
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| Simplify. | |
| Check: Substitute 1 for . |
TRY IT 7.1
Solve: .
Show answer
q = 1
TRY IT 7.2
Solve: .
Show answer
n = 1
EXAMPLE 8
Solve: .
Solution
This equation can be solved following the process in previous examples. But let us also look at an alternate way to solve it.
This equation has on the left and on the right. Since > , make the right side the variable side and the left side the constant side.
| Subtract from both sides to remove the variable term from the left. | |
| Combine like terms. | |
| Subtract 8 from both sides to remove the constant from the right. | |
| Simplify. | |
| Divide both sides by 3 to make 1 the coefficient of . | |
| Simplify. | |
| Check: Let . |
Note that we could have made the left side the variable side instead of the right side, but it would have led to a negative coefficient on the variable term. While we could work with the negative, there is less chance of error when working with positives. The strategy outlined above helps avoid the negatives!
TRY IT 8.1
Solve: .
Show answer
a = −5
TRY IT 8.2
Solve: .
Show answer
k = −6
To solve an equation with fractions, we still follow the same steps to get the solution.
EXAMPLE 9
Solve: .
Solution
Since > , make the left side the variable side and the right side the constant side.
| Subtract from both sides. | |
| Combine like terms. | |
| Subtract 5 from both sides. | |
| Simplify. | |
| Check: Let . |
TRY IT 9.1
Solve: .
Show answer
x = 10
TRY IT 9.2
Solve: .
Show answer
y = −3
We follow the same steps when the equation has decimals, too.
EXAMPLE 10
Solve: .
Solution
Since > , make the left side the variable side and the right side the constant side.
| Subtract from both sides. | |
| Combine like terms. | |
| Subtract 4 from both sides. | |
| Simplify. | |
| Use the Division Property of Equality. | |
| Simplify. | |
| Check: Let . |
TRY IT 10.1
Solve: .
Show answer
x = −5
TRY IT 10.2
Solve: .
Show answer
y = −5
Solve Equations Using a General Strategy
Each of the first few sections of this chapter has dealt with solving one specific form of a linear equation. It’s time now to lay out an overall strategy that can be used to solve any linear equation. We call this the general strategy. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.
HOW TO: Use a General Strategy for Solving Linear Equations
- Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
- Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
- Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
- Make the coefficient of the variable term to equal to . Use the Multiplication or Division Property of Equality. State the solution to the equation.
- Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.
EXAMPLE 11
Solve: .
Solution
| Simplify each side of the equation as much as possible. Use the Distributive Property. |
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| Collect all variable terms on one side of the equation—all s are already on the left side. | |
| Collect constant terms on the other side of the equation. Subtract 6 from each side |
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| Simplify. | |
| Make the coefficient of the variable term equal to 1. Divide each side by 3. | |
| Simplify. | |
| Check: Let . |
TRY IT 11.1
Solve: .
Show answer
x = 4
TRY IT 11.2
Solve: .
Show answer
y = 1
EXAMPLE 12
Solve: .
Solution
| Simplify each side of the equation as much as possible by distributing. The only term is on the left side, so all variable terms are on the left side of the equation. |
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| Add 5 to both sides to get all constant terms on the right side of the equation. | |
| Simplify. | |
| Make the coefficient of the variable term equal to 1 by multiplying (or dividing) both sides by -1. |
OR |
| Simplify. | |
| Check: Let . |
TRY IT 12.1
Solve: .
Show answer
y = −6
TRY IT 12.2
Solve: .
Show answer
z = 8
EXAMPLE 13
Solve: .
Solution
| Simplify each side of the equation as much as possible. Distribute. |
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| Combine like terms | |
| The only is on the left side, so all variable terms are on one side of the equation. | |
| Add 3 to both sides to get all constant terms on the other side of the equation. | |
| Simplify. | |
| Make the coefficient of the variable term equal to 1 by dividing both sides by 4. | |
| Simplify. | |
| Check: Let . |
TRY IT 13.1
Solve: .
Show answer
a = 2
TRY IT 13.2
Solve: .
Show answer
n = 2
EXAMPLE 14
Solve: .
Solution
Be careful when distributing the negative.
| Simplify—use the Distributive Property. |
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| Combine like terms. | |
| Add 2 to both sides to collect constants on the right. | |
| Simplify. | |
| Divide both sides by −6. | |
| Simplify. | |
| Check: Let . |
TRY IT 14.1
Solve: .
Show answer
TRY IT 14.2
Solve: .
Show answer
EXAMPLE 15
Solve: .
Solution
| Distribute. | |
| Combine like terms. | |
| Subtract to get all the variables on the right since > . | |
| Simplify. | |
| Subtract 9 to get the constants on the left. | |
| Simplify. | |
| Divide by 5. | |
| Simplify. | |
| Check: Substitute: . |
TRY IT 15.1
Solve: .
Show answer
p = −2
TRY IT 15.2
Solve: .
Show answer
q = −8
EXAMPLE 16
Solve: .
Solution
| Distribute. |
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| Add to get all the variables on the left. | |
| Simplify. | |
| Add 1 to get constants on the right. | |
| Simplify. | |
| Divide by 4. | |
| Simplify. | |
| Check: Let . |
TRY IT 16.1
Solve: .
Show answer
u = 2
TRY IT 16.2
Solve: .
Show answer
x = 4
In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.
EXAMPLE 17
Solve: .
Solution
| Distribute. | |
| Subtract to get all the s to the left. | |
| Simplify. | |
| Subtract 1.2 to get the constants to the right. | |
| Simplify. | |
| Divide. | |
| Simplify. | |
| Check: Let . |
TRY IT 17.1
Solve: .
Show answer
1
TRY IT 17.2
Solve: .
Show answer
−1
Key Concepts
- Solve an equation with variables and constants on both sides
- Choose one side to be the variable side and then the other will be the constant side.
- Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
- Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
- Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
- Check the solution by substituting into the original equation.
- General strategy for solving linear equations
- Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
- Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
- Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
- Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation.
- Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.
Practice Makes Perfect
Solve an Equation with Constants on Both Sides
In the following exercises, solve the equation for the variable.
| 1. | 2. |
| 3. | 4. |
| 5. | 6. |
| 7. | 8. |
| 9. | 10. |
| 11. | 12. |
Solve an Equation with Variables on Both Sides
In the following exercises, solve the equation for the variable.
| 13. | 14. |
| 15. | 16. |
| 17. | 18. |
| 19. | 20. |
| 21. | 22. |
| 23. | 24. |
Solve an Equation with Variables and Constants on Both Sides
In the following exercises, solve the equations for the variable.
| 25. | 26. |
| 27. | 28. |
| 29. | 30. |
| 31. | 32. |
| 33. | 34. |
| 35. | 36. |
| 37. | 38. |
| 39. | 40. |
| 41. | 42. |
| 43. | 44. |
| 45. | 46. |
| 47. | 48. |
| 49. | 50. |
Solve an Equation Using the General Strategy
In the following exercises, solve the linear equation using the general strategy.
| 51. | 52. |
| 53. | 54. |
| 55. | 56. |
| 57. | 58. |
| 59. | 60. |
| 61. | 62. |
| 63. | 64. |
| 65. | 66. |
| 67. | 68. |
| 69. | 70. |
| 71. | 72. |
| 73. | 74. |
| 75. | 76. |
| 77. | 78. |
| 79. | 80. |
| 81. | 82. |
| 83. | 84. |
| 85. | 86. |
Everyday Math
| Making a fence 87. Jovani has a fence around the rectangular garden in his backyard. The perimeter of the fence is feet. The length is feet more than the width. Find the width, , by solving the equation . | Concert tickets 88. At a school concert, the total value of tickets sold was Student tickets sold for and adult tickets sold for The number of adult tickets sold was less than times the number of student tickets. Find the number of student tickets sold, , by solving the equation . |
| Coins 89. Rhonda has in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the number of nickels, , by solving the equation . | Fencing 90. Micah has feet of fencing to make a rectangular dog pen in his yard. He wants the length to be feet more than the width. Find the length, , by solving the equation . |
Writing Exercises
| 91. When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient as the variable side? | 92. Solve the equation , explaining all the steps of your solution. |
| 93. What is the first step you take when solving the equation Explain why this is your first step. | 94. Solve the equation explaining all the steps of your solution as in the examples in this section. |
| 95. Using your own words, list the steps in the General Strategy for Solving Linear Equations. | 96. 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. |
Answers
| 1. | 6 | 3. | 6 | 5. | -8 | 7. | -8 |
| 9. | -4 | 11. | -2 | 13. | -11 | 15. | 9 |
| 17. | -3 | 19. | 3 | 21. | 23. | -2 | |
| 25. | 19 | 27. | 7 | 29. | -5 | 31. | -4 |
| 33. | 2 | 35. | 4 | 37. | -6 | 39. | 7 |
| 41. | -40 | 43. | 15 | 45. | 3.46 | 47. | 60 |
| 49. | 23 | 51. | 9 | 53. | 6 | 55. | 3 |
| 57. | -2 | 59. | -1 | 61. | 5 | 63. | 1 |
| 65. | -4 | 67. | -9 | 69. | 2 | 71. | 6 |
| 73. | 75. | 3 | 77. | -4 | 79. | 2 | |
| 81. | 34 | 83. | 10 | 85. | 87. | 30 feet | |
| 89. | 8 nickels | 91. | Answers will vary. | 93. | Answers will vary. | 95. | Answers will vary. |
Attributions
This chapter has been adapted from “Solve Equations with Variables and Constants on Both Sides” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.