Graphing Systems of Linear Inequalities

Lynn Marecek

41 Graphing Systems of Linear Inequalities

Learning Objectives

By the end of this section, you will be able to:

Determine whether an ordered pair is a solution of a system of linear inequalities
Solve a system of linear inequalities by graphing
Solve applications of systems of inequalities

Before you get started, take this readiness quiz.

Solve the inequality $2a<5a+12.$

If you missed this problem, review (Figure).
Determine whether the ordered pair $\left(3,\frac{1}{2}\right)$ is a solution to the system $y>2x+3.$

If you missed this problem, review (Figure).

Determine whether an ordered pair is a solution of a system of linear inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

System of Linear Inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.

$\left\{\begin{array}{c}x+4y\ge 10\hfill \\ 3x-2y<12\hfill \end{array}$

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs $\left(x,y\right)$ that make both inequalities true.

Solutions of a System of Linear Inequalities

Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.

The solution of a system of linear inequalities is shown as a shaded region in the x, y coordinate system that includes all the points whose ordered pairs make the inequalities true.

To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

Determine whether the ordered pair is a solution to the system $\left\{\begin{array}{c}x+4y\ge 10\hfill \\ 3x-2y<12\hfill \end{array}.$

ⓐ $\left(-2,4\right)$ ⓑ $\left(3,1\right)$

ⓐ Is the ordered pair $\left(-2,4\right)$ a solution?

The ordered pair $\left(-2,4\right)$ made both inequalities true. Therefore $\left(-2,4\right)$ is a solution to this system.

ⓑ Is the ordered pair $\left(3,1\right)$ a solution?

The ordered pair $\left(3,1\right)$ made one inequality true, but the other one false. Therefore $\left(3,1\right)$ is not a solution to this system.

Determine whether the ordered pair is a solution to the system: $\left\{\begin{array}{c}x-5y>10\hfill \\ 2x+3y>-2\hfill \end{array}.$

ⓐ $\left(3,-1\right)$ ⓑ $\left(6,-3\right)$

ⓐ no ⓑ yes

Determine whether the ordered pair is a solution to the system: $\left\{\begin{array}{c}y>4x-2\hfill \\ 4x-y<20\hfill \end{array}.$

ⓐ $\left(-2,1\right)$ ⓑ $\left(4,-1\right)$

ⓐ yes ⓑ no

Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

How to Solve a System of Linear Inequalities by Graphing

Solve the system by graphing: $\left\{\begin{array}{c}y\ge 2x-1\hfill \\ y<x+1\hfill \end{array}.$

Step 1. Graph the first inequality. We graph y less than 2x minus 1. Graph the boundary line y equal to 2x minus 1. It is a solid line because the inequality sign is less than. Shade in the side of the boundary line where the inequality is true. We choose 0, 0 as a test point. It is a solution to the equation, so we shade in above the boundary line.

Step 2. On the same grid, graph the second inequality y less than x plus 1. Graph the boundary line y equal to x plus 1. It is a dashed line because the inequality sign is less than. Shade in the side of that boundary line where the inequality is true. Again, we use 0, 0 as a test point. It is a solution so we shade below the line y equals x plus 1.

Step 3. The solution is the region where the shading overlaps. The point where the boundary lines intersect is not a solution because it is not a solution to y less than x plus 1. The solution is all points in the area bounded by the lines on the bottom left.

Step 4. Check by choosing a test point. We use minus 1, minus 1. Substituting in the inequality y less than 2x minus 1, we get minus 1 less than minus 3 which is true. Hence, it is a solution. Similarly, it is also true for the other inequality. The region containing minus 1, minus 1 is the solution to this system.

Solve the system by graphing: $\left\{\begin{array}{c}y<3x+2\hfill \\ y>\text{−}x-1\hfill \end{array}.$

The figure shows a graph plotted for the inequalities y less than three times x plus two and y greater than minus x minus one. Two lines intersect each other on the graph. An area to the right of both the lines is colored in grey. It is the solution.

The solution is the grey region.

Solve the system by graphing: $\left\{\begin{array}{c}y<-\frac{1}{2}x+3\hfill \\ y<3x-4\hfill \end{array}.$

The figure shows the graph plotted for the inequalities y less than minus half of x plus three and y less than three times x minus four. Two intersecting lines are shown on the graph. The area bound by the two lines to the bottom right is shown in grey. It is the solution.

The solution is the grey region.

Solve a system of linear inequalities by graphing.

Graph the first inequality.
- Graph the boundary line.
- Shade in the side of the boundary line where the inequality is true.
On the same grid, graph the second inequality.
- Graph the boundary line.
- Shade in the side of that boundary line where the inequality is true.
The solution is the region where the shading overlaps.
Check by choosing a test point.

Solve the system by graphing: $\left\{\begin{array}{c}x-y>3\hfill \\ y<-\frac{1}{5}x+4\hfill \end{array}.$

$\left\{\begin{array}{c}x-y>3\hfill \\ y<-\frac{1}{5}x+4\hfill \end{array}$

Graph x − y > 3, by graphing x − y = 3

and testing a point.

The intercepts are x = 3 and y = −3 and the

boundary line will be dashed.

Test (0, 0) which makes the inequality false so shade

(red) the side that does not contain (0, 0).

Graph $y<-\frac{1}{5}x+4$ by graphing $y=-\frac{1}{5}x+4$

using the slope $m=-\frac{1}{5}$ and y-intercept b = 4.

The boundary line will be dashed

Test (0, 0) which makes the inequality true, so

shade (blue) the side that contains (0, 0).

Choose a test point in the solution and verify that it is a solution to both inequalties.

The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice—which appears as the darkest shaded region.

Solve the system by graphing: $\left\{\begin{array}{c}x+y\le 2\hfill \\ y\ge \frac{2}{3}x-1\hfill \end{array}.$

The figure shows the graph for the inequalities x plus y less than or equal to two and y greater than or equal to two by three of x minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.

The solution is the grey region.

Solve the system by graphing: $\left\{\begin{array}{c}3x-2y\le 6\hfill \\ y>\text{−}\frac{1}{4}x+5\hfill \end{array}.$

The figure shows graph for the inequalities three times x minus two times y less than or equal to six and y greater than or equal to minus one by four of x plus five. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.

The solution is the grey region.

Solve the system by graphing: $\left\{\begin{array}{c}x-2y<5\hfill \\ y>-4\hfill \end{array}.$

$\phantom{\rule{2em}{0ex}}\left\{\begin{array}{c}x-2y<5\hfill \\ y>-4\hfill \end{array}$

Graph $x-2y<5,$ by graphing $x-2y=5$

and testing a point. The intercepts are

x = 5 and y = −2.5 and the

boundary line will be dashed.

Test (0, 0) which makes the inequality true, so shade

(red) the side that contains (0, 0).

Graph $y>-4,$ by graphing $y=-4$ and

recognizing that it is a horizontal line

through $y=-4.$ The boundary line will

be dashed.

Test (0, 0) which makes the inequality

true so shade (blue) the side that contains (0, 0).

The point $\left(0,0\right)$ is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed.

The solution is the area shaded twice—which appears as the darkest shaded region.

Solve the system by graphing: $\left\{\begin{array}{c}y\ge 3x-2\hfill \\ y<-1\hfill \end{array}.$

The figure shows graph for the inequalities y greater than or equal to three times x minus two and y less than minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution

The solution is the grey region.

Solve the system by graphing: $\left\{\begin{array}{c}x>-4\hfill \\ x-2y\ge -4\hfill \end{array}.$

The figure shows graph for the inequalities x greater than or equal to minus four and x minus two times y greater than minus four. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.

The solution is the grey region.

Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in the next example.

Solve the system by graphing: $\left\{\begin{array}{c}4x+3y\ge 12\hfill \\ y<-\frac{4}{3}x+1\hfill \end{array}.$

$\left\{\begin{array}{c}4x+3y\ge 12\hfill \\ y<-\frac{4}{3}x+1\hfill \end{array}$

Graph $4x+3y\ge 12,$ by graphing $4x+3y=12$

and testing a point. The intercepts are x = 3

and y = 4 and the boundary line will be solid.

Test (0, 0) which makes the inequality false, so

shade (red) the side that does not contain (0, 0).

Graph $y<-\frac{4}{3}x+1$ by graphing $y=-\frac{4}{3}x+1$

using the slope $m=-\frac{4}{3}$ and y-intercept

b = 1. The boundary line will be dashed.

Test (0, 0) which makes the inequality true, so

shade (blue) the side that contains (0, 0).

There is no point in both shaded regions, so the system has no solution.

Solve the system by graphing: $\left\{\begin{array}{c}3x-2y\ge 12\hfill \\ y\ge \frac{3}{2}x+1\hfill \end{array}.$

The graph of three times x minus two times y greater than or equal to twelve and y greater than or equal to three by two of x plus one is shown. Two intersecting lines are shown. The inequalities do not have a solution.

No solution.

Solve the system by graphing: $\left\{\begin{array}{c}x+3y>8\hfill \\ y<-\frac{1}{3}x-2\hfill \end{array}.$

The graph of x plus three times y greater than eight and y less than minus one by three of x minus two is shown. Two intersecting lines are shown. The inequalities do not have a solution.

No solution.

Some systems of linear inequalities where the boundary lines are parallel will have a solution. We’ll see this in the next example.

Solve the system by graphing: $\left\{\begin{array}{c}y>\frac{1}{2}x-4\hfill \\ x-2y<-4\hfill \end{array}.$

$\phantom{\rule{2em}{0ex}}\left\{\begin{array}{c}y>\frac{1}{2}x-4\hfill \\ x-2y<-4\hfill \end{array}$

Graph $y>\frac{1}{2}x-4$ by graphing $y=\frac{1}{2}x-4$

using the slope $m=\frac{1}{2}$ and the intercept

b = −4. The boundary line will be dashed.

Test (0, 0) which makes the inequality true, so

shade (red) the side that contains (0, 0).

Graph $x-2y<-4$ by graphing $x-2y=-4$

and testing a point. The intercepts are

x = −4 and y = 2 and the boundary line will be dashed.

Choose a test point in the solution and verify

that it is a solution to both inequalties.

Test (0, 0) which makes the inequality false, so

shade (blue) the side that does not contain (0, 0).

No point on the boundary lines is included in the solution as both lines are dashed.

The solution is the region that is shaded twice which is also the solution to $x-2y<-4.$

Solve the system by graphing: $\left\{\begin{array}{c}y\ge 3x+1\hfill \\ -3x+y\ge -4\hfill \end{array}.$

The figure shows the graph of the inequalities y greater than or equal to three times x plus one and minus three times x plus y greater than or equal to minus four. Two parallel lines are shown and the region to the left of both is colored in grey. It is the solution.

The solution is the grey region.

Solve the system by graphing: $\left\{\begin{array}{c}y\le -\frac{1}{4}x+2\hfill \\ x+4y\le 4\hfill \end{array}.$

The figure shows the graph of the inequalities y less than or equal to minus one fourth of x plus 2 and x plus four times y less than or equal to four. Two parallel lines are shown and the region to the bottom of both is colored in grey. It is the solution.

The solution is the grey region.

Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so we add inequalities to the system as additional requirements.

Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her ?4 and each large photo costs her ?10. She doesn’t want to spend more than ?200 on photos to display.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Could she display 20 large and 10 small photos?

ⓐ

$\begin{array}{cc}\text{Let}\hfill & x=\text{the number of small photos}.\hfill \\ & y=\text{the number of large photos}\hfill \end{array}$

To find the system of equations translate the information.

$\begin{array}{}\\ \\ \\ \phantom{\rule{3em}{0ex}}\text{She wants to have at least 25 photos.}\hfill \\ \phantom{\rule{3em}{0ex}}\text{The number of small plus the number of large should be at least 25.}\hfill \\ \hfill x+y\ge 25\hfill \\ \\ \\ \phantom{\rule{3em}{0ex}}\text{?4 for each small and ?10 for each large must be no more than ?200}\hfill \\ \hfill 4x+10y\le 200\hfill \\ \\ \\ \phantom{\rule{3em}{0ex}}\text{The number of small photos must be greater than or equal to 0.}\hfill \\ \hfill x\ge 0\hfill \\ \\ \\ \phantom{\rule{3em}{0ex}}\text{The number of large photos must be greater than or equal to 0.}\hfill \\ \hfill y\ge 0\hfill \end{array}$

We have our system of equations.

$\phantom{\rule{1em}{0ex}}\left\{\begin{array}{c}x+y\ge 25\hfill \\ 4x+10y\le 200\hfill \\ x\ge 0\hfill \\ y\ge 0\hfill \end{array}$

ⓑ

Since $x\ge 0$ and $y\ge 0$ (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.

To graph $x+y\ge 25,$ graph $x+y=25$ as a solid line.

Choose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0).

To graph $4x+10y\le 200,$ graph $4x+10y=200$ as a solid line.

Choose (0, 0) as a test point. Since it does make the inequality true, shade (blue) the side that include the point (0, 0).

The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly-shaded section are included in the solution as are the points on the x-axis from (25, 0) to (55, 0).

ⓒ To determine if 10 small and 20 large photos would work, we look at the graph to see if the point (10, 20) is in the solution region. We could also test the point to see if it is a solution of both equations.

It is not, Christy would not display 10 small and 20 large photos.

ⓓ To determine if 20 small and 10 large photos would work, we look at the graph to see if the point (20, 10) is in the solution region. We could also test the point to see if it is a solution of both equations.

It is, so Christy could choose to display 20 small and 10 large photos.

Notice that we could also test the possible solutions by substituting the values into each inequality.

A trailer can carry a maximum weight of 160 pounds and a maximum volume of 15 cubic feet. A microwave oven weighs 30 pounds and has 2 cubic feet of volume, while a printer weighs 20 pounds and has 3 cubic feet of space.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Could 7 microwaves and 3 printers be carried on this trailer?

ⓐ $\left\{\begin{array}{c}30m+20p\le 160\hfill \\ 2m+3p\le 15\hfill \end{array}$

ⓑ

The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.

ⓓ no

Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost ?2 and the answer sheets cost ?1. Mary’s budget for these supplies allows for a maximum cost of ?400.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Could Mary purchase 150 pencils and 150 answer sheets?

ⓐ $\left\{\begin{array}{c}a\ge p+5\hfill \\ a+2p\le 400\hfill \end{array}$

ⓑ

The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.

ⓓ no

When we use variables other than x and y to define an unknown quantity, we must change the names of the axes of the graph as well.

Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn’t want to spend more than ?5. At the hamburger restaurant near his college, each hamburger has 240 calories and costs ?1.40. Each cookie has 160 calories and costs ?0.50.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Could he eat 2 hamburgers and 4 cookies?

ⓐ

$\begin{array}{cc}\text{Let}\hfill & h=\text{the number of hamburgers}.\hfill \\ & c=\text{the number of cookies}\hfill \end{array}$

To find the system of equations translate the information.

The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be more that 800.

$\begin{array}{c}\hfill 240h+160c\ge 800\hfill \\ \\ \\ \phantom{\rule{3em}{0ex}}\text{The amount spent on hamburgers at ?1.40 each, plus the amount spent on cookies}\hfill \\ \phantom{\rule{3em}{0ex}}\text{at ?0.50 each must be no more than ?5.00.}\hfill \\ \hfill 1.40h+0.50c\le 5\hfill \\ \\ \\ \phantom{\rule{3em}{0ex}}\text{The number of hamburgers must be greater than or equal to 0.}\hfill \\ \hfill h\ge 0\hfill \\ \\ \\ \phantom{\rule{3em}{0ex}}\text{The number of cookies must be greater than or equal to 0.}\hfill \\ \hfill c\ge 0\hfill \end{array}$

$\begin{array}{cccc}\text{We have our system of equations.}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\left\{\begin{array}{c}240h+160c\ge 800\hfill \\ 1.40h+0.50c\le 5\hfill \\ h\ge 0\hfill \\ c\ge 0\hfill \end{array}\hfill \end{array}$

ⓑ

Since $h>=0$ and $c>=0$ (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.

To graph $240h+160c\ge 800,$ graph $240h+160c=800$ as a solid line.

Choose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0).

Graph $1.40h+0.50c\le 5.$ The boundary line is $1.40h+0.50c=5.$ We test (0, 0) and it makes the inequality true. We shade the side of the line that includes (0, 0).

The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly shaded section are included in the solution as are the points on the x-axis from (5, 0) to (10, 0).

ⓓ To determine if 2 hamburgers and 4 cookies would meet Omar’s criteria, we see if the point (2, 4) is in the solution region. It is, Omar might choose to eat 2 hamburgers and 4 cookies.

We could also test the possible solutions by substituting the values into each inequality.

Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only ?25 to spend on the extra food he needs and will spend it on ?0.75 donuts which have 360 calories each and ?2 energy drinks which have 110 calories.

ⓐ Write a system of inequalities that models this situation.

ⓑ Graph the system.

ⓓ Can he buy 1 donut and 3 energy drinks and satisfy his caloric needs?

ⓐ $\left\{\begin{array}{c}0.75d+2e\le 25\hfill \\ 360d+110e\ge 1000\hfill \end{array}$

ⓑ

The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.

ⓓ no

Philip’s doctor tells him he should add at least 1,000 more calories per day to his usual diet. Philip wants to buy protein bars that cost ?1.80 each and have 140 calories and juice that costs ?1.25 per bottle and have 125 calories. He doesn’t want to spend more than ?12.

ⓐ Write a system of inequalities that models this situation.

ⓑ Graph the system.

ⓓ Can he buy 5 protein bars and 3 bottles of juice?

ⓐ $\left\{\begin{array}{c}140p+125j\ge 1000\hfill \\ 1.80p+1.25j\le 12\hfill \end{array}$

ⓑ

The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.

ⓓ no

Access these online resources for additional instruction and practice with solving systems of linear inequalities by graphing.

Key Concepts

Solutions of a System of Linear Inequalities: Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the x, y coordinate system that includes all the points whose ordered pairs make the inequalities true.
How to solve a system of linear inequalities by graphing.
1. Graph the first inequality.
  
  Graph the boundary line.
  
  Shade in the side of the boundary line where the inequality is true.
2. On the same grid, graph the second inequality.
  
  Graph the boundary line.
  
  Shade in the side of that boundary line where the inequality is true.
3. The solution is the region where the shading overlaps.
4. Check by choosing a test point.

Section Exercises

Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

$\left\{\begin{array}{c}3x+y>5\hfill \\ 2x-y\le 10\hfill \end{array}$

ⓐ $\left(3,-3\right)$

ⓑ $\left(7,1\right)$

$\left\{\begin{array}{c}4x-y<10\hfill \\ -2x+2y>-8\hfill \end{array}$

ⓐ $\left(5,-2\right)$

ⓑ $\left(-1,3\right)$

ⓐ false ⓑ true

$\left\{\begin{array}{c}y>\frac{2}{3}x-5\hfill \\ x+\frac{1}{2}y\le 4\hfill \end{array}$

ⓐ $\text{(6, −4)}$

ⓑ $\text{(3, 0)}$

$\left\{\begin{array}{c}y<\frac{3}{2}x+3\hfill \\ \frac{3}{4}x-2y<5\hfill \end{array}$

ⓐ $\left(-4,-1\right)$

ⓑ $\text{(8, 3)}$

ⓐ false ⓑ true

$\left\{\begin{array}{c}7x+2y>14\hfill \\ 5x-y\le 8\hfill \end{array}$

ⓐ $\text{(2, 3)}$

ⓑ $\text{(7, −1)}$

$\left\{\begin{array}{c}6x-5y<20\hfill \\ -2x+7y>-8\hfill \end{array}$

ⓐ $\text{(1, −3)}$

ⓑ $\text{(−4, 4)}$

ⓐ false ⓑ true

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

$\left\{\begin{array}{c}y\le 3x+2\hfill \\ y>x-1\hfill \end{array}$

$\left\{\begin{array}{c}y<-2x+2\hfill \\ y\ge -x-1\hfill \end{array}$

The figure shows the graph of inequalities y less than minus two times x plus two and y greater than or equal to minus x minus one. Two intersecting lines are shown, one in red and the other in blue. The area bound by the two lines is shown in grey.

The solution is the grey region.

$\left\{\begin{array}{c}y<2x-1\hfill \\ y\le -\frac{1}{2}x+4\hfill \end{array}$

$\left\{\begin{array}{c}y\ge -\frac{2}{3}x+2\hfill \\ y>2x-3\hfill \end{array}$

The figure shows the graph of the inequalities y greater than or equal to minus two by three x plus two and y greater than two times x minus three. Two intersecting lines, one in red and the other in blue, are shown. The region bound by them is shown in grey.

The solution is the grey region.

$\left\{\begin{array}{c}x-y>1\hfill \\ y<-\frac{1}{4}x+\hfill \end{array}3$

$\left\{\begin{array}{c}x+2y<4\hfill \\ y<x-2\hfill \end{array}$

The figure shows the graph of the inequalities x minus two times y less than four and y less than x minus two. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey.

The solution is the grey region.

$\left\{\begin{array}{c}3x-y\ge 6\hfill \\ y\ge -\frac{1}{2}x\hfill \end{array}$

$\left\{\begin{array}{c}2x+4y\ge 8\hfill \\ y\le \frac{3}{4}x\hfill \end{array}$

The figure shows the graph of the inequalities two times x plus four times y greater than or equal to eight and y less than or equal to minus three fourth of x. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

$\left\{\begin{array}{c}2x-5y<10\hfill \\ 3x+4y\ge 12\hfill \end{array}$

$\left\{\begin{array}{c}3x-2y\le 6\hfill \\ -4x-2y>8\hfill \end{array}$

The figure shows the graph of the inequalities three times x minus two times y less than or equal to six and minus four times x minus two times y greater than eight. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

$\left\{\begin{array}{c}2x+2y>-4\hfill \\ \text{−}x+3y\ge 9\hfill \end{array}$

$\left\{\begin{array}{c}2x+y>-6\hfill \\ \text{−}x+2y\ge -4\hfill \end{array}$

The figure shows the graph of the inequalities two times x plus y greater than minus six and minus x plus two times y greater than or equal to minus four. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

$\left\{\begin{array}{c}x-2y<3\hfill \\ y\le 1\hfill \end{array}$

$\left\{\begin{array}{c}x-3y>4\hfill \\ y\le -1\hfill \end{array}$

The figure shows the graph of the inequalities x minus three times y greater than four and y less than or equal to minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

$\left\{\begin{array}{c}y\ge -\frac{1}{2}x-3\hfill \\ x\le 2\hfill \end{array}$

$\left\{\begin{array}{c}y\le -\frac{2}{3}x+5\hfill \\ x\ge 3\hfill \end{array}$

The figure shows the graph of the inequality y less than or equal to minus two by three times x plus five and x greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

$\left\{\begin{array}{c}y\ge \frac{3}{4}x-2\hfill \\ y<2\hfill \end{array}$

$\left\{\begin{array}{c}y\le -\frac{1}{2}x+3\hfill \\ y<1\hfill \end{array}$

The figure shows the graph of the inequalities y less than or equal to minus half x plus three and y less than one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

$\left\{\begin{array}{c}3x-4y<8\hfill \\ x<1\hfill \end{array}$

$\left\{\begin{array}{c}-3x+5y>10\hfill \\ x>-1\hfill \end{array}$

The figure shows the graph of the inequalities minus three times x plus five times y greater than ten and x greater than minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

$\left\{\begin{array}{c}x\ge 3\hfill \\ y\le 2\hfill \end{array}$

$\left\{\begin{array}{c}x\le -1\hfill \\ y\ge 3\hfill \end{array}$

The figure shows the graph of the inequalities x less than or equal to minus one and y greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

$\left\{\begin{array}{c}2x+4y>4\hfill \\ y\le -\frac{1}{2}x-2\hfill \end{array}$

$\left\{\begin{array}{c}x-3y\ge 6\hfill \\ y>\frac{1}{3}x+1\hfill \end{array}$

The figure shows the graph of the inequalities x minus three times y greater than or equal to six and y greater than one third of x plus one. Two non intersecting lines, one in blue and the other in red, are shown.

No solution.

$\left\{\begin{array}{c}-2x+6y<0\hfill \\ 6y>2x+4\hfill \end{array}$

$\left\{\begin{array}{c}-3x+6y>12\hfill \\ 4y\le 2x-4\hfill \end{array}$

The figure shows the graph of the inequalities minus three times x plus six times y greater than twelve and four times y less than or equal to two times x minus four. Two non intersecting lines, one in blue and the other in red, are shown.

No solution.

$\left\{\begin{array}{c}y\ge -3x+2\hfill \\ 3x+y>5\hfill \end{array}$

$\left\{\begin{array}{c}y\ge \frac{1}{2}x-1\hfill \\ -2x+4y\ge 4\hfill \end{array}$

The figure shows the graph of the inequalities y greater than or equal to minus half x minus one and minus two times x plus four times y greater than or equal to four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

The solution is the grey region.

$\left\{\begin{array}{c}y\le -\frac{1}{4}x-2\hfill \\ x+4y<6\hfill \end{array}$

$\left\{\begin{array}{c}y\ge 3x-1\hfill \\ -3x+y>-4\hfill \end{array}$

The figure shows the graph of the inequalities y greater than or equal to three times x minus one and minus three times x plus y greater than minus four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

The solution is the grey region.

$\left\{\begin{array}{c}3y>x+2\hfill \\ -2x+6y>8\hfill \end{array}$

$\left\{\begin{array}{c}y<\frac{3}{4}x-2\hfill \\ -3x+4y<7\hfill \end{array}$

The figure shows the graph of the inequalities y less than three by fourth x minus two and minus three x plus four y less than seven. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

The solution is the grey region.

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for ?15 and the landscapes for ?10. She needs to sell at least ?800 worth of drawings in order to earn a profit.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Will she make a profit if she sells 50 portraits and 20 landscapes?

Jake does not want to spend more than ?50 on bags of fertilizer and peat moss for his garden. Fertilizer costs ?2 a bag and peat moss costs ?5 a bag. Jake’s van can hold at most 20 bags.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Can he buy 10 bags of fertilizer and 10 bags of peat moss?

ⓐ $\left\{\begin{array}{c}f\ge 0\hfill \\ p\ge 0\hfill \\ f+p\le 20\hfill \\ 2f+5p\le 50\hfill \end{array}$

ⓑ

The figure shows the graph of the inequalities f plus p less than or equal to twenty and two f and five p less than or equal to fifty. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

ⓓ no

Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than ?500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is ?3 and for a package the cost is ?7.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Can she mail 90 cards and 40 packages?

Juan is studying for his final exams in chemistry and algebra. he knows he only has 24 hours to study, and it will take him at least three times as long to study for algebra than chemistry.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Can he spend 6 hours on chemistry and 18 hours on algebra?

ⓐ $\left\{\begin{array}{c}c\ge 0\hfill \\ a\ge 0\hfill \\ c+a\le 24\hfill \\ a\ge 3c\hfill \end{array}$

ⓑ

The figure shows the graph of the inequalities c plus a less than or equal to twenty four and a greater than or equal to three times c. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

ⓓ no

Jocelyn is pregnant and so she needs to eat at least 500 more calories a day than usual. When buying groceries one day with a budget of ?15 for the extra food, she buys bananas that have 90 calories each and chocolate granola bars that have 150 calories each. The bananas cost ?0.35 each and the granola bars cost ?2.50 each.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Could she buy 3 bananas and 4 granola bars?

Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs ?3.20 and a protein bar costs ?1.75. The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has ?10 dollars to spend

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Could he buy no bottles of protein water and 5 protein bars?

ⓐ $\left\{\begin{array}{c}w\ge 0\hfill \\ b\ge 0\hfill \\ 27w+16b>80\hfill \\ 3.20w+1.75b\le 10\hfill \end{array}$

ⓑ

The figure shows the graph of the inequalities twenty seven times w plus sixteen times b greater than eighty and three point two times w plus one point seven five b less than or equal to ten. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

ⓓ yes

Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?

Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum of 1500 calories from this exercise. Walking burns 270 calories/mile and running burns 650 calories.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Could he his goal by walking 2 miles and running 2 mile

ⓐ $\left\{\begin{array}{c}w\ge 0\hfill \\ r\ge 0\hfill \\ w+r\ge 4\hfill \\ 270w+650r\ge 1500\hfill \end{array}$

ⓑ

The figure shows the graph of the inequalities w plus r greater than or equals to four and two seventy w plus six fifty r greater than or equal to fifteen hundred. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

ⓓ yes

Writing Exercises

Graph the inequality $x-y\ge 3.$ How do you know which side of the line $x-y=3$ should be shaded?

Graph the system $\left\{\begin{array}{c}x+2y\le 6\hfill \\ y\ge -\frac{1}{2}x-4\hfill \end{array}.$ What does the solution mean?

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Chapter Review Exercises

Solve Systems of Linear Equations with Two Variables

Determine Whether an Ordered Pair is a Solution of a System of Equations.

In the following exercises, determine if the following points are solutions to the given system of equations.

$\left\{\begin{array}{c}x+3y=-9\hfill \\ 2x-4y=12\hfill \end{array}$

ⓐ $\left(-3,-2\right)$

ⓑ $\left(0,-3\right)$

$\left\{\begin{array}{c}x+y=8\hfill \\ y=x-4\hfill \end{array}$

ⓐ $\left(6,2\right)$

ⓑ $\left(9,-1\right)$

ⓐ yes ⓑ no

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

$\left\{\begin{array}{c}3x+y=6\hfill \\ x+3y=-6\hfill \end{array}$

$\left\{\begin{array}{c}x+4y=-1\hfill \\ x=3\hfill \end{array}$

The figure shows the graph of equations x plus four times y equal to minus one and x equal to three. Two intersecting lines are shown.

$\left(3,-1\right)$

$\left\{\begin{array}{c}2x-y=5\hfill \\ 4x-2y=10\hfill \end{array}$

$\left\{\begin{array}{c}\text{−}x+2y=4\hfill \\ y=\frac{1}{2}x-3\hfill \end{array}$

The figure shows the graph for the equations minus x plus two times y equal to four and y equal to half x minus three. Two parallel lines are shown.

no solution

In the following exercises, without graphing determine the number of solutions and then classify the system of equations.

$\left\{\begin{array}{c}y=\frac{2}{5}x+2\hfill \\ -2x+5y=10\hfill \end{array}$

$\left\{\begin{array}{c}3x+2y=6\hfill \\ y=-3x+4\hfill \end{array}$

one solution, consistent system, independent equations

$\left\{\begin{array}{c}5x-4y=0\hfill \\ y=\frac{5}{4}x-5\hfill \end{array}$

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

$\left\{\begin{array}{c}3x-2y=2\hfill \\ y=\frac{1}{2}x+3\hfill \end{array}$

$\left(4,5\right)$

$\left\{\begin{array}{c}x-y=0\hfill \\ 2x+5y=-14\hfill \end{array}$

$\left\{\begin{array}{c}y=-2x+7\hfill \\ y=\frac{2}{3}x-1\hfill \end{array}$

$\left(3,1\right)$

$\left\{\begin{array}{c}y=-5x\hfill \\ 5x+y=6\hfill \end{array}$

$\left\{\begin{array}{c}y=-\frac{1}{3}x+2\hfill \\ x+3y=6\hfill \end{array}$

infinitely many solutions

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination

$\left\{\begin{array}{c}x+y=12\hfill \\ x-y=-10\hfill \end{array}$

$\left\{\begin{array}{c}3x-8y=20\hfill \\ x+3y=1\hfill \end{array}$

$\left(4,-1\right)$

$\left\{\begin{array}{c}9x+4y=2\hfill \\ 5x+3y=5\hfill \end{array}$

$\left\{\begin{array}{c}\frac{1}{3}x-\frac{1}{2}y=1\hfill \\ \frac{3}{4}x-y=\frac{5}{2}\hfill \end{array}$

$\left(6,2\right)$

$\left\{\begin{array}{c}\text{−}x+3y=8\hfill \\ 2x-6y=-20\hfill \end{array}$

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

$\left\{\begin{array}{c}6x-5y=27\hfill \\ 3x+10y=-24\hfill \end{array}$

elimination

$\left\{\begin{array}{c}y=3x-9\hfill \\ 4x-5y=23\hfill \end{array}$

Solve Applications with Systems of Equations

Solve Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Mollie wants to plant 200 bulbs in her garden, all irises and tulips. She wants to plant three times as many tulips as irises. How many irises and how many tulips should she plant?

50 irises and 150 tulips

Ashanti has been offered positions by two phone companies. The first company pays a salary of ?22,000 plus a commission of ?100 for each contract sold. The second pays a salary of ?28,000 plus a commission of ?25 for each contract sold. How many contract would need to be sold to make the total pay the same?

Leroy spent 20 minutes jogging and 40 minutes cycling and burned 600 calories. The next day, Leroy swapped times, doing 40 minutes of jogging and 20 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?

10 calories jogging and 10 calories cycling

Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and calculator. Troy bought four notebooks and five calculators for ?116. Lisa bought two notebooks and three calculators for ?68. Find the cost of each notebook and each thumb drive.

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The difference of two supplementary angles is 58 degrees. Find the measures of the angles.

119, 61

Two angles are complementary. The measure of the larger angle is five more than four times the measure of the smaller angle. Find the measures of both angles.

The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

$35\text{°}$ and $55\text{°}$

Becca is hanging a 28 foot floral garland on the two sides and top of a pergola to prepare for a wedding. The height is four feet less than the width. Find the height and width of the pergola.

The perimeter of a city rectangular park is 1428 feet. The length is 78 feet more than twice the width. Find the length and width of the park.

the length is 450 feet, the width is 264 feet

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

Sheila and Lenore were driving to their grandmother’s house. Lenore left one hour after Sheila. Sheila drove at a rate of 45 mph, and Lenore drove at a rate of 60 mph. How long will it take for Lenore to catch up to Sheila?

Bob left home, riding his bike at a rate of 10 miles per hour to go to the lake. Cheryl, his wife, left 45 minutes $\left(\frac{3}{4}$ hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?

$\frac{1}{2}$ an hour

Marcus can drive his boat 36 miles down the river in three hours but takes four hours to return upstream. Find the rate of the boat in still water and the rate of the current.

A passenger jet can fly 804 miles in 2 hours with a tailwind but only 776 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

the rate of the jet is 395 mph, the rate of the wind is 7 mph

Solve Mixture Applications with Systems of Equations

Solve Mixture Applications with Systems of Equations

For the following exercises, translate to a system of equations and solve.

Lynn paid a total of ?2,780 for 261 tickets to the theater. Student tickets cost ?10 and adult tickets cost ?15. How many student tickets and how many adult tickets did Lynn buy?

Priam has dimes and pennies in a cup holder in his car. The total value of the coins is ?4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?

41 dimes and 11 pennies

Yumi wants to make 12 cups of party mix using candies and nuts. Her budget requires the party mix to cost her ?1.29 per cup. The candies are ?2.49 per cup and the nuts are ?0.69 per cup. How many cups of candies and how many cups of nuts should she use?

A scientist needs 70 liters of a 40% solution of alcohol. He has a 30% and a 60% solution available. How many liters of the 30% and how many liters of the 60% solutions should he mix to make the 40% solution?

$46\frac{2}{3}$ liters of 30% solution, $23\frac{1}{3}$ liters of 60% solution

Solve Interest Applications

For the following exercises, translate to a system of equations and solve.

Jack has ?12,000 to invest and wants to earn 7.5% interest per year. He will put some of the money into a savings account that earns 4% per year and the rest into CD account that earns 9% per year. How much money should he put into each account?

When she graduates college, Linda will owe ?43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was ?1,585. What is the amount of each loan?

?29,000 for the federal loan, ?14,000 for the private loan

Solve Systems of Equations with Three Variables

Solve Systems of Equations with Three Variables

In the following exercises, determine whether the ordered triple is a solution to the system.

$\left\{\begin{array}{c}3x-4y-3z=2\hfill \\ 2x-6y+z=3\hfill \\ 2x+3y-2z=3\hfill \end{array}$

ⓐ $\left(2,3,-1\right)$

ⓑ $\left(3,1,3\right)$

$\left\{\begin{array}{c}y=\frac{2}{3}x-2\hfill \\ x+3y-z=15\hfill \\ x-3y+z=-2\hfill \end{array}$

ⓐ $\left(-6,5,\frac{1}{2}\right)$

ⓑ $\left(5,\frac{4}{3},-3\right)$

ⓐ no ⓑ yes

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

$\left\{\begin{array}{c}3x-5y+4z=5\hfill \\ 5x+2y+z=0\hfill \\ 2x+3y-2z=3\hfill \end{array}$

$\left\{\begin{array}{c}x+\frac{5}{2}y+z=-2\hfill \\ 2x+2y+\frac{1}{2}z=-4\hfill \\ \frac{1}{3}x-y-z=1\hfill \end{array}$

$\left(-3,2,-4\right)$

$\left\{\begin{array}{c}5x+3y=-6\hfill \\ 2y+3z=-1\hfill \\ 7x+z=1\hfill \end{array}$

$\left\{\begin{array}{c}2x+3y+z=12\hfill \\ x+y+z=9\hfill \\ 3x+4y+2z=20\hfill \end{array}$

no solution

$\left\{\begin{array}{c}\text{−}x-3y+2z=14\hfill \\ \text{−}x+2y-3z=-4\hfill \\ 3x+y-2z=6\hfill \end{array}$

Solve Applications using Systems of Linear Equations with Three Variables

After attending a major league baseball game, the patrons often purchase souvenirs. If a family purchases 4 t-shirts, a cap and 1 stuffed animal their total is ?135. A couple buys 2 t-shirts, a cap and 3 stuffed animals for their nieces and spends ?115. Another couple buys 2 t-shirts, a cap and 1 stuffed animal and their total is ?85. What is the cost of each item?

25, 20, 15

Solve Systems of Equations Using Matrices

Write the Augmented Matrix for a System of Equations.

Write each system of linear equations as an augmented matrix.

$\left\{\begin{array}{c}3x-y=-1\hfill \\ -2x+2y=5\hfill \end{array}$

$\left\{\begin{array}{c}4x+3y=-2\hfill \\ x-2y-3z=7\hfill \\ 2x-y+2z=-6\hfill \end{array}$

$\left[\begin{array}{cccccccccc}\hfill 4& & & \hfill 3& & & \hfill 0& & & \hfill -2\\ \hfill 1& & & \hfill -2& & & \hfill -3& & & \hfill 7\\ \hfill 2& & & \hfill -1& & & \hfill 2& & & \hfill -6\end{array}\right]$

Write the system of equations that that corresponds to the augmented matrix.

$\left[\begin{array}{cccc}2\hfill & & & \hfill -4\\ 3\hfill & & & \hfill -3\end{array}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill -2\\ \hfill -1\end{array}\right]$

$\left[\begin{array}{ccccccc}\hfill 1& & & \hfill 0& & & \hfill -3\\ \hfill 1& & & \hfill -2& & & \hfill 0\\ \hfill 0& & & \hfill -1& & & \hfill 2\end{array}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill -1\\ \hfill -2\\ \hfill 3\end{array}\right]$

$\left\{\begin{array}{c}x-3z=-1\hfill \\ x-2y=-27\hfill \\ \text{−}y+2z=3\hfill \end{array}$

In the following exercises, perform the indicated operations on the augmented matrices.

$\left[\begin{array}{cccc}4\hfill & & & \hfill -6\\ 3\hfill & & & \hfill 2\end{array}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill -3\\ \hfill 1\end{array}\right]$

ⓐ Interchange rows 2 and 1.

ⓑ Multiply row 1 by 4.

$\left[\begin{array}{ccccccc}1\hfill & & & \hfill -3& & & \hfill -2\\ 2\hfill & & & \hfill 2& & & \hfill -1\\ 4\hfill & & & \hfill -2& & & \hfill -3\end{array}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill 4\\ \hfill -3\\ \hfill -1\end{array}\right]$

ⓐ Interchange rows 2 and 3.

ⓑ Multiply row 1 by 2.

ⓐ $\left[\begin{array}{cccccccccc}1\hfill & & & \hfill -3& & & \hfill -2& & & \hfill 4\\ 4\hfill & & & \hfill -2& & & \hfill -3& & & \hfill -1\\ 2\hfill & & & \hfill 2& & & \hfill -1& & & \hfill -3\end{array}\right]$

ⓑ $\left[\begin{array}{cccccccccc}2\hfill & & & \hfill -6& & & \hfill -4& & & \hfill 8\\ 4\hfill & & & \hfill -2& & & \hfill -3& & & \hfill -1\\ 2\hfill & & & \hfill 2& & & \hfill -1& & & \hfill -3\end{array}\right]$

ⓒ $\left[\begin{array}{cccccccccc}2\hfill & & & \hfill -6& & & \hfill -4& & & \hfill 8\\ 4\hfill & & & \hfill -2& & & \hfill -3& & & \hfill -1\\ 0\hfill & & & \hfill -6& & & \hfill -1& & & \hfill 5\end{array}\right]$

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

$\left\{\begin{array}{c}4x+y=6\hfill \\ x-y=4\hfill \end{array}$

$\left\{\begin{array}{c}2x-y+3z=-3\hfill \\ \text{−}x+2y-z=10\hfill \\ x+y+z=5\hfill \end{array}$

$\left(-2,5,-2\right)$

$\left\{\begin{array}{c}2y+3z=-1\hfill \\ 5x+3y=-6\hfill \\ 7x+z=1\hfill \end{array}$

$\left\{\begin{array}{c}x+2y-3z=-1\hfill \\ x-3y+z=1\hfill \\ 2x-y-2z=2\hfill \end{array}$

no solution

$\left\{\begin{array}{c}x+y-3z=-1\hfill \\ y-z=0\hfill \\ \text{−}x+2y=1\hfill \end{array}$

Solve Systems of Equations Using Determinants

Evaluate the Determinant of a 2 × 2 Matrix

In the following exercise, evaluate the determinate of the square matrix.

$\left[\begin{array}{cccc}8\hfill & & & \hfill -4\\ 5\hfill & & & \hfill -3\end{array}\right]$

$-4$

Evaluate the Determinant of a 3 × 3 Matrix

In the following exercise, find and then evaluate the indicated minors.

$|\begin{array}{ccccccc}\hfill -1& & & \hfill -3& & & \hfill 2\\ \hfill 4& & & \hfill -2& & & \hfill -1\\ \hfill -2& & & \hfill 0& & & \hfill -3\end{array}|;$ Find the minor ⓐ ${a}_{1}$ ⓑ ${b}_{1}$ ⓒ ${c}_{2}$

In the following exercise, evaluate each determinant by expanding by minors along the first row.

$|\begin{array}{ccccccc}\hfill -2& & & \hfill -3& & & \hfill -4\\ \hfill 5& & & \hfill -6& & & \hfill 7\\ \hfill -1& & & \hfill 2& & & \hfill 0\end{array}|$

21

In the following exercise, evaluate each determinant by expanding by minors.

$|\begin{array}{ccccccc}\hfill 3& & & \hfill 5& & & \hfill 4\\ \hfill -1& & & \hfill 3& & & \hfill 0\\ \hfill -2& & & \hfill 6& & & \hfill 1\end{array}|$

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s rule

$\left\{\begin{array}{c}x-3y=-9\hfill \\ 2x+5y=4\hfill \end{array}$

$\left(-3,2\right)$

$\left\{\begin{array}{c}4x-3y+z=7\hfill \\ 2x-5y-4z=3\hfill \\ 3x-2y-2z=-7\hfill \end{array}$

$\left\{\begin{array}{c}2x+5y=4\hfill \\ 3y-z=3\hfill \\ 4x+3z=-3\hfill \end{array}$

$\left(-3,2,3\right)$

$\left\{\begin{array}{c}x+y-3z=-1\hfill \\ y-z=0\hfill \\ \text{−}x+2y=1\hfill \end{array}$

$\left\{\begin{array}{c}3x+4y-3z=-2\hfill \\ 2x+3y-z=-12\hfill \\ x+y-2z=6\hfill \end{array}$

inconsistent

Solve Applications Using Determinants

In the following exercises, determine whether the given points are collinear.

$\left(0,2\right),$ $\left(-1,-1\right),$ and $\left(-2,4\right)$

Graphing Systems of Linear Inequalities

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

$\left\{\begin{array}{c}4x+y>6\hfill \\ 3x-y\le 12\hfill \end{array}$

ⓐ $\left(2,-1\right)$

ⓑ $\left(3,-2\right)$

ⓐ yes ⓑ no

$\left\{\begin{array}{c}y>\frac{1}{3}x+2\hfill \\ x-\frac{1}{4}y\le 10\hfill \end{array}$

ⓐ $\left(6,5\right)$

ⓑ $\left(15,8\right)$

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

$\left\{\begin{array}{c}y<3x+1\hfill \\ y\ge -x-2\hfill \end{array}$

The figure shows the graph of inequalities y less than three times x plus one and y greater than or equal to minus x minus two. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.

The solution is the grey region.

$\left\{\begin{array}{c}x-y>-1\hfill \\ y<\frac{1}{3}x-2\hfill \end{array}$

$\left\{\begin{array}{c}2x-3y<6\hfill \\ 3x+4y\ge 12\hfill \end{array}$

The figure shows the graph of inequalities two times x minus three times y less six and three times x plus four times y greater than or equal to twelve. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.

The solution is the grey region.

$\left\{\begin{array}{c}y\le -\frac{3}{4}x+1\hfill \\ x\ge -5\hfill \end{array}$

$\left\{\begin{array}{c}x+3y<5\hfill \\ y\ge -\frac{1}{3}x+6\hfill \end{array}$

The figure shows the graph of inequalities x plus three times y less than five and y greater than or equal to minus one third x plus six. Two parallel lines, one in red and the other in blue, are shown. An area is shown in grey.

No solution.

$\left\{\begin{array}{c}y\ge 2x-5\hfill \\ -6x+3y>-4\hfill \end{array}$

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Roxana makes bracelets and necklaces and sells them at the farmers’ market. She sells the bracelets for ?12 each and the necklaces for ?18 each. At the market next weekend she will have room to display no more than 40 pieces, and she needs to sell at least ?500 worth in order to earn a profit.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Should she display 39 bracelets and 1 necklace?

ⓐ $\left\{\begin{array}{c}b\ge 0\hfill \\ n\ge 0\hfill \\ b+n\le 40\hfill \\ 12b+18n\ge 500\hfill \end{array}$

ⓑ

The figure shows the graph of b plus n equal to forty and twelve b plus eighteen n equal to five hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.

ⓓ no

Annie has a budget of ?600 to purchase paperback books and hardcover books for her classroom. She wants the number of hardcover to be at least 5 more than three times the number of paperback books. Paperback books cost ?4 each and hardcover books cost ?15 each.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓓ Can she buy 10 paperback books and 37 hardcover books?

Chapter Practice Test

In the following exercises, solve the following systems by graphing.

$\left\{\begin{array}{c}x-y=5\hfill \\ x+2y=-4\hfill \end{array}$

The figure shows the graph of inequalities h equal to three p plus five and four times p plus fifteen times h equal to six hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.

$\left(2,-3\right)$

$\left\{\begin{array}{c}x-y>-2\hfill \\ y\le 3x+1\hfill \end{array}$

In the following exercises, solve each system of equations. Use either substitution or elimination.

$\left\{\begin{array}{c}x+4y=6\hfill \\ -2x+y=-3\hfill \end{array}$

$\left(2,1\right)$

$\left\{\begin{array}{c}-3x+4y=25\hfill \\ x-5y=-23\hfill \end{array}$

$\left\{\begin{array}{c}x+y-z=-1\hfill \\ 2x-y+2z=8\hfill \\ -3x+2y+z=-9\hfill \end{array}$

$\left(2,-2,1\right)$

Solve the system of equations using a matrix.

$\left\{\begin{array}{c}2x+y=7\hfill \\ x-2y=6\hfill \end{array}$

$\left\{\begin{array}{c}-3x+y+z=-4\hfill \\ \text{−}x+2y-2z=1\hfill \\ 2x-y-z=-1\hfill \end{array}$

$\left(5,7,4\right)$

Solve using Cramer’s rule.

$\left\{\begin{array}{c}3x+y=-3\hfill \\ 2x+3y=6\hfill \end{array}$

Evaluate the determinant by expanding by minors:

$|\begin{array}{ccccccc}\hfill 3& & & \hfill -2& & & \hfill -2\\ \hfill 2& & & \hfill -1& & & \hfill 4\\ \hfill -1& & & \hfill 0& & & \hfill -3\end{array}|.$

$9$

In the following exercises, translate to a system of equations and solve.

Greg is paddling his canoe upstream, against the current, to a fishing spot 10 miles away. If he paddles upstream for 2.5 hours and his return trip takes 1.25 hours, find the speed of the current and his paddling speed in still water.

A pharmacist needs 20 liters of a 2% saline solution. He has a 1% and a 5% solution available. How many liters of the 1% and how many liters of the 5% solutions should she mix to make the 2% solution?

15 liters of 1% solution, 5 liters of 5% solution

Arnold invested ?64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received ?4,500 in interest in one year?

The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of ?65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of ?140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 can of popcorn for a total sales of ?250. What is the cost of each item?

The candy cost ?20; the cookies cost ?5; and the popcorn cost ?10.

The manufacturer of a granola bar spends ?1.20 to make each bar and sells them for ?2. The manufacturer also has fixed costs each month of ?8,000.

ⓐ Find the cost function C when x granola bars are manufactured

ⓑ Find the revenue function R when x granola bars are sold.

ⓓ Find the break-even point. Interpret what the break-even point means.

Translate to a system of inequalities and solve.

Andi wants to spend no more than ?50 on Halloween treats. She wants to buy candy bars that cost ?1 each and lollipops that cost ?0.50 each, and she wants the number of lollipops to be at least three times the number of candy bars.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓐ $\left\{\begin{array}{c}C\ge 0\hfill \\ L\ge 0\hfill \\ C+0.5L\le 50\hfill \\ L\ge 3C\hfill \end{array}$

ⓑ

The figure shows the graph of two equations. Two intersecting lines, one in red and the other in blue, are shown. The red line passes through origin. An area is shown in grey.

ⓓ yes

Glossary

system of linear inequalities: Two or more linear inequalities grouped together form a system of linear inequalities.

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Learning Objectives

Determine whether an ordered pair is a solution of a system of linear inequalities

Solve a System of Linear Inequalities by Graphing

Solve Applications of Systems of Inequalities

Key Concepts

Section Exercises

Practice Makes Perfect

Writing Exercises

Self Check

Chapter Review Exercises

Chapter Practice Test

Glossary

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