4.7 Graphing Systems of Linear Inequalities

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.

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.

${\begin{matrix} x + 4 y \geq 10 \\ 3 x - 2 y < 12 \end{matrix}$

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 $(x, y)$ that make both 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.

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.

Example 3

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

	${\begin{matrix} x - y > 3 \\ y < - \frac{1}{5} x + 4 \end{matrix}$
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.

A Cartesian coordinate graph displays two linear inequalities. The area above the dashed blue line is shaded light blue, representing one solution set. The area below the dashed red line is shaded dark grey, representing another solution set.

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.

Example 4

Solve the system by graphing: ${ x minus 2 y less than 5; y greater than −4.$

	${ x minus 2 y less than 5; y greater than −4$
Graph $x minus 2 y less than 5,$ by graphing $x - 2 y = 5$ 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 greater than −4,

by graphing

y equals −4

and
recognizing that it is a horizontal line
through

y equals −4.

The boundary line will
be dashed.

Test (0, 0) which makes the inequality
true so shade (blue) the side that contains (0, 0).

Graph of a system of linear inequalities with two dashed lines, y=1/2x-3 and y=-4, showing three distinct shaded regions: gray, light blue, and light orange.

The point $open parenthesis 0, 0 close parenthesis$ 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.

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

Example 5

Solve the system by graphing: ${\begin{matrix} 4 x + 3 y \geq 12 \\ y < - \frac{4}{3} x + 1 \end{matrix} .$

	${\begin{matrix} 4 x + 3 y \geq 12 \\ y < - \frac{4}{3} x + 1 \end{matrix}$
Graph $4 x plus 3 y greater than or equal to 12,$ by graphing $4 x + 3 y = 12$ 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.

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

Example 6

Solve the system by graphing: ${\begin{matrix} y > \frac{1}{2} x - 4 \\ x - 2 y < −4 \end{matrix} .$

	${\begin{matrix} y > \frac{1}{2} x - 4 \\ x - 2 y < −4 \end{matrix}$
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 $x minus 2 y less than −4$ by graphing $x - 2 y = −4$ 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 $x minus 2 y less than −4.$

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.

Example 7

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 10 small and 20 large photos?
ⓓ Could she display 20 small and 10 large photos?

ⓐ

Let

x = the number of small photos .

y = the number of large photos

To find the system of equations translate the information.

She wants to have at least 25 photos.

The number of small plus the number of large should be at least 25.

x plus y greater than or equal to 25

$4 for each small and $10 for each large must be no more than $200

4 x plus 10 y less than or equal to 200

The number of small photos must be greater than or equal to 0.

x greater than or equal to 0

The number of large photos must be greater than or equal to 0.

y greater than or equal to 0

We have our system of equations.

{\begin{matrix} x + y \geq 25 \\ 4 x + 10 y \leq 200 \\ x \geq 0 \\ y \geq 0 \end{matrix}

ⓑ
Since $x greater than or equal to 0$ and $y greater than or equal to 0$ (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 plus y greater than or equal to 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

4 x plus 10 y less than or equal to 200,

graph

4 x + 10 y = 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).

A graph on a coordinate plane shows two downward-sloping lines representing linear inequalities. Individual shaded regions highlight their solutions, with a darker overlapping area indicating the feasible region.

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.

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.

Example 8

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 3 hamburgers and 1 cookie?
ⓓ Could he eat 2 hamburgers and 4 cookies?

ⓐ

Let

h = the number of hamburgers .

c = the number of cookies

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.

240 h plus 160 c greater than or equal to 800

The amount spent on hamburgers at $1.40 each, plus the amount spent on cookies at $0.50 each

must be no more than $5.00.

1.40 h plus 0.50 c less than or equal to 5

The number of hamburgers must be greater than or equal to 0.

h greater than or equal to 0

The number of cookies must be greater than or equal to 0.

c greater than or equal to 0

We have our system of equations.

{\begin{matrix} 240 h + 160 c \geq 800 \\ 1.40 h + 0.50 c \leq 5 \\ h \geq 0 \\ c \geq 0 \end{matrix}

ⓑ
Since $h greater than equals 0$ and $c greater than equals 0$ (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

240 h plus 160 c greater than or equal to 800,

graph

240 h + 160 c = 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).

A graph on an h-c coordinate plane shows two intersecting lines and three distinct shaded regions, representing solutions to a system of inequalities.

Graph $1.40 h plus 0.50 c less than or equal to 5.$ The boundary line is $1.40 h + 0.50 c = 5 .$ 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.

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.

${\begin{matrix} 3 x + y > 5 \\ 2 x - y \leq 10 \end{matrix}$

ⓐ $open parenthesis 3, −3 close parenthesis$ ⓑ $open parenthesis 7, 1 close parenthesis$

${\begin{matrix} 4 x - y < 10 \\ −2 x + 2 y > −8 \end{matrix}$

ⓐ $open parenthesis 5, −2 close parenthesis$ ⓑ $open parenthesis −1, 3 close parenthesis$

ⓐ false ⓑ true

${\begin{matrix} y > \frac{2}{3} x - 5 \\ x + \frac{1}{2} y \leq 4 \end{matrix}$

ⓐ $(6, −4)$ ⓑ $(3, 0)$

${\begin{matrix} y < \frac{3}{2} x + 3 \\ \frac{3}{4} x - 2 y < 5 \end{matrix}$

ⓐ $open parenthesis −4, −1 close parenthesis$ ⓑ $(8, 3)$

ⓐ false ⓑ true

${\begin{matrix} 7 x + 2 y > 14 \\ 5 x - y \leq 8 \end{matrix}$

ⓐ $(2, 3)$ ⓑ $(7, −1)$

${\begin{matrix} 6 x - 5 y < 20 \\ −2 x + 7 y > −8 \end{matrix}$

ⓐ $(1, −3)$ ⓑ $(−4, 4)$

ⓐ false ⓑ true

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y \leq 3 x + 2 \\ y > x - 1 \end{matrix}$

${\begin{matrix} y < - 2 x + 2 \\ y \geq - x - 1 \end{matrix}$

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.

${\begin{matrix} y < 2 x - 1 \\ y \leq - \frac{1}{2} x + 4 \end{matrix}$

${\begin{matrix} y \geq - \frac{2}{3} x + 2 \\ y > 2 x - 3 \end{matrix}$

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.

$\{\begin{matrix} x - y > 1 \\ y < - \frac{1}{4} x + 3 \end{matrix}$

${\begin{matrix} x + 2 y < 4 \\ y < x - 2 \end{matrix}$

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.

${\begin{matrix} 3 x - y \geq 6 \\ y \geq - \frac{1}{2} x \end{matrix}$

${\begin{matrix} 2 x + 4 y \geq 8 \\ y \leq \frac{3}{4} x \end{matrix}$

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.

${\begin{matrix} 2 x - 5 y < 10 \\ 3 x + 4 y \geq 12 \end{matrix}$

${\begin{matrix} 3 x - 2 y \leq 6 \\ −4 x - 2 y > 8 \end{matrix}$

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.

${\begin{matrix} 2 x + 2 y > −4 \\ − x + 3 y \geq 9 \end{matrix}$

${\begin{matrix} 2 x + y > −6 \\ − x + 2 y \geq −4 \end{matrix}$

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.

${ x minus 2 y less than 3; y less than or equal to 1$

${\begin{matrix} x - 3 y > 4 \\ y \leq - 1 \end{matrix}$

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.

${\begin{matrix} y \geq - \frac{1}{2} x - 3 \\ x \leq 2 \end{matrix}$

${\begin{matrix} y \leq - \frac{2}{3} x + 5 \\ x \geq 3 \end{matrix}$

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.

${\begin{matrix} y \geq \frac{3}{4} x - 2 \\ y < 2 \end{matrix}$

${\begin{matrix} y \leq - \frac{1}{2} x + 3 \\ y < 1 \end{matrix}$

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.

${ 3 x minus 4 y less than 8; x less than 1$

${ −3 x plus 5 y greater than 10; x greater than −1$

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.

${ x greater than or equal to 3; y less than or equal to 2$

${ x less than or equal to −1; y greater than or equal to 3$

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.

${\begin{matrix} 2 x + 4 y > 4 \\ y \leq - \frac{1}{2} x - 2 \end{matrix}$

${\begin{matrix} x - 3 y \geq 6 \\ y > \frac{1}{3} x + 1 \end{matrix}$

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.

${\begin{matrix} −2 x + 6 y < 0 \\ 6 y > 2 x + 4 \end{matrix}$

${\begin{matrix} −3 x + 6 y > 12 \\ 4 y \leq 2 x - 4 \end{matrix}$

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.

${\begin{matrix} y \geq −3 x + 2 \\ 3 x + y > 5 \end{matrix}$

${\begin{matrix} y \geq \frac{1}{2} x - 1 \\ −2 x + 4 y \geq 4 \end{matrix}$

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.

${\begin{matrix} y \leq - \frac{1}{4} x - 2 \\ x + 4 y < 6 \end{matrix}$

${\begin{matrix} y \geq 3 x - 1 \\ −3 x + y > −4 \end{matrix}$

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.

${\begin{matrix} 3 y > x + 2 \\ −2 x + 6 y > 8 \end{matrix}$

${\begin{matrix} y < \frac{3}{4} x - 2 \\ −3 x + 4 y < 7 \end{matrix}$

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 20 portraits and 35 landscapes?
ⓓ 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 15 bags of fertilizer and 4 bags of peat moss?
ⓓ Can he buy 10 bags of fertilizer and 10 bags of peat moss?

ⓐ ${\begin{matrix} f \geq 0 \\ p \geq 0 \\ f + p \leq 20 \\ 2 f + 5 p \leq 50 \end{matrix}$
ⓑ

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.

ⓒ yes
ⓓ 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 60 cards and 26 packages?
ⓓ 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 4 hours on chemistry and 20 hours on algebra?
ⓓ Can he spend 6 hours on chemistry and 18 hours on algebra?

ⓐ ${ c greater than or equal to 0; a greater than or equal to 0; c plus a less than or equal to 24; a greater than or equal to 3 c$
ⓑ

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.

ⓒ yes
ⓓ 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 5 bananas and 6 granola bars?
ⓓ 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 3 bottles of protein water and 1 protein bar?
ⓓ Could he buy no bottles of protein water and 5 protein bars?

ⓐ ${\begin{matrix} w \geq 0 \\ b \geq 0 \\ 27 w + 16 b > 80 \\ 3.20 w + 1.75 b \leq 10 \end{matrix}$
ⓑ

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.

ⓒ no
ⓓ 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 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?
ⓓ 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 meet his goal by walking 3 miles and running 1 mile?
ⓓ Could he meet his goal by walking 2 miles and running 2 miles?

ⓐ ${\begin{matrix} w \geq 0 \\ r \geq 0 \\ w + r \geq 4 \\ 270 w + 650 r \geq 1500 \end{matrix}$
ⓑ

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.

ⓒ no
ⓓ yes

Writing Exercises

Graph the inequality $x minus y greater than or equal to 3.$ How do you know which side of the line $x - y = 3$ should be shaded?

Graph the system ${\begin{matrix} x + 2 y \leq 6 \\ y \geq - \frac{1}{2} x - 4 \end{matrix} .$ 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.

The figure shows a table with four columns and four rows. The first row is the title row. The four titles are I can…, confidently, with some help and No – I don’t get it! In the second row of the first column, the text says ‘determine whether an ordered pair is a solution of a system of linear inequalities.’ In the third row of the first column, the text says ‘solve applications of systems of inequalities.’

ⓑ 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.

${\begin{matrix} x + 3 y = −9 \\ 2 x - 4 y = 12 \end{matrix}$
ⓐ $open parenthesis −3, −2 close parenthesis$
ⓑ $open parenthesis 0, −3 close parenthesis$

${\begin{matrix} x + y = 8 \\ y = x - 4 \end{matrix}$
ⓐ $open parenthesis 6, 2 close parenthesis$
ⓑ $open parenthesis 9, −1 close parenthesis$

ⓐ yes ⓑ no

Solve a System of Linear Equations by Graphing

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

${\begin{matrix} 3 x + y = 6 \\ x + 3 y = −6 \end{matrix}$

${\begin{matrix} x + 4 y = −1 \\ x = 3 \end{matrix}$

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.

open parenthesis 3, −1 close parenthesis

${\begin{matrix} 2 x - y = 5 \\ 4 x - 2 y = 10 \end{matrix}$

${\begin{matrix} − x + 2 y = 4 \\ y = \frac{1}{2} x - 3 \end{matrix}$

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.

${\begin{matrix} y = \frac{2}{5} x + 2 \\ −2 x + 5 y = 10 \end{matrix}$

${\begin{matrix} 3 x + 2 y = 6 \\ y = −3 x + 4 \end{matrix}$

one solution, consistent system, independent equations

${\begin{matrix} 5 x - 4 y = 0 \\ y = \frac{5}{4} x - 5 \end{matrix}$

Solve a System of Equations by Substitution

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

${\begin{matrix} 3 x - 2 y = 2 \\ y = \frac{1}{2} x + 3 \end{matrix}$

$open parenthesis 4, 5 close parenthesis$

${\begin{matrix} x - y = 0 \\ 2 x + 5 y = −14 \end{matrix}$

${\begin{matrix} y = −2 x + 7 \\ y = \frac{2}{3} x - 1 \end{matrix}$

$open parenthesis 3, 1 close parenthesis$

${\begin{matrix} y = −5 x \\ 5 x + y = 6 \end{matrix}$

${\begin{matrix} y = - \frac{1}{3} x + 2 \\ x + 3 y = 6 \end{matrix}$

infinitely many solutions

Solve a System of Equations by Elimination

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

${\begin{matrix} x + y = 12 \\ x - y = −10 \end{matrix}$

${\begin{matrix} 3 x - 8 y = 20 \\ x + 3 y = 1 \end{matrix}$

$open parenthesis 4, −1 close parenthesis$

${\begin{matrix} 9 x + 4 y = 2 \\ 5 x + 3 y = 5 \end{matrix}$

${\begin{matrix} \frac{1}{3} x - \frac{1}{2} y = 1 \\ \frac{3}{4} x - y = \frac{5}{2} \end{matrix}$

$open parenthesis 6, 2 close parenthesis$

${\begin{matrix} − x + 3 y = 8 \\ 2 x - 6 y = −20 \end{matrix}$

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.

${\begin{matrix} 6 x - 5 y = 27 \\ 3 x + 10 y = −24 \end{matrix}$

elimination

${\begin{matrix} y = 3 x - 9 \\ 4 x - 5 y = 23 \end{matrix}$

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 thumb drive. Troy bought four notebooks and five thumb drives for $116. Lisa bought two notebooks and three thumb drives 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 degrees and 61 degrees

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 degrees and 55 degrees

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.

Length = 502 feet, Width = 212 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 $(\frac{3}{4}$ 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.

${\begin{matrix} 3 x - 4 y - 3 z = 2 \\ 2 x - 6 y + z = 3 \\ 2 x + 3 y - 2 z = 3 \end{matrix}$
ⓐ $open parenthesis 2, 3, −1 close parenthesis$
ⓑ $open parenthesis 3, 1, 3 close parenthesis$

${\begin{matrix} y = \frac{2}{3} x - 2 \\ x + 3 y - z = 15 \\ x - 3 y + z = −2 \end{matrix}$
ⓐ $(−6, 5, \frac{1}{2})$
ⓑ $(5, \frac{4}{3}, −3)$

ⓐ no ⓑ no

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} 3 x - 5 y + 4 z = 5 \\ 5 x + 2 y + z = 0 \\ 2 x + 3 y - 2 z = 3 \end{matrix}$

${\begin{matrix} x + \frac{5}{2} y + z = −2 \\ 2 x + 2 y + \frac{1}{2} z = −4 \\ \frac{1}{3} x - y - z = 1 \end{matrix}$

$open parenthesis −3, 2, −4 close parenthesis$

${\begin{matrix} 5 x + 3 y = −6 \\ 2 y + 3 z = −1 \\ 7 x + z = 1 \end{matrix}$

${\begin{matrix} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{matrix}$

no solution

${\begin{matrix} − x - 3 y + 2 z = 14 \\ − x + 2 y - 3 z = −4 \\ 3 x + y - 2 z = 6 \end{matrix}$

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.

${\begin{matrix} 3 x - y = −1 \\ −2 x + 2 y = 5 \end{matrix}$

${\begin{matrix} 4 x + 3 y = −2 \\ x - 2 y - 3 z = 7 \\ 2 x - y + 2 z = −6 \end{matrix}$

$[ 4, 3, 0, −2; 1, −2, −3, 7; 2, −1, 2, −6 ]$

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

$[ 2, −4; 3, −3 | −2; −1 ]$

$[ 1, 0, −3; 1, −2, 0; 0, −1, 2 | −1; −2; 3 ]$

${\begin{matrix} x - 3 z = −1 \\ x - 2 y = −2 \\ − y + 2 z = 3 \end{matrix}$

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

$[ 4, −6; 3, 2 | −3; 1 ]$

$[ 1, −3, −2; 2, 2, −1; 4, −2, −3 | 4; −3; −1 ]$

ⓐ $[ 1, −3, −2, 4; 4, −2, −3, −1; 2, 2, −1, −3 ]$
ⓑ $[ 2, −6, −4, 8; 4, −2, −3, −1; 2, 2, −1, −3 ]$
ⓒ $[ 2, −6, −4, 8; 4, −2, −3, −1; 0, −6, −1, 5 ]$

Solve Systems of Equations Using Matrices

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

${\begin{matrix} 4 x + y = 6 \\ x - y = 4 \end{matrix}$

${\begin{matrix} 2 x - y + 3 z = −3 \\ − x + 2 y - z = 10 \\ x + y + z = 5 \end{matrix}$

$open parenthesis −2, 5, 2 close parenthesis$

${\begin{matrix} 2 y + 3 z = −1 \\ 5 x + 3 y = −6 \\ 7 x + z = 1 \end{matrix}$

${\begin{matrix} x + 2 y - 3 z = −1 \\ x - 3 y + z = 1 \\ 2 x - y - 2 z = 2 \end{matrix}$

no solution

${\begin{matrix} x + y - 3 z = −1 \\ y - z = 0 \\ − x + 2 y = 1 \end{matrix}$

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.

$[ 8, −4; 5, −3 ]$

$−4$

Evaluate the Determinant of a 3 × 3 Matrix

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

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

$| −2, −3, −4; 5, −6, 7; −1, 2, 0 |$

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

$| 3, 5, 4; −1, 3, 0; −2, 6, 1 |$

Use Cramer’s Rule to Solve Systems of Equations

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

${\begin{matrix} x - 3 y = −9 \\ 2 x + 5 y = 4 \end{matrix}$

$open parenthesis −3, 2 close parenthesis$

${\begin{matrix} 4 x - 3 y + z = 7 \\ 2 x - 5 y - 4 z = 3 \\ 3 x - 2 y - 2 z = −7 \end{matrix}$

${\begin{matrix} 2 x + 5 y = 4 \\ 3 y - z = 3 \\ 4 x + 3 z = −3 \end{matrix}$

$open parenthesis −3, 2, 3 close parenthesis$

${\begin{matrix} x + y - 3 z = −1 \\ y - z = 0 \\ − x + 2 y = 1 \end{matrix}$

${\begin{matrix} 3 x + 4 y - 3 z = −2 \\ 2 x + 3 y - z = −12 \\ x + y - 2 z = 6 \end{matrix}$

inconsistent

Solve Applications Using Determinants

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

$open parenthesis 0, 2 close parenthesis,$ $open parenthesis −1, −1 close parenthesis,$ and $open parenthesis −2, 4 close parenthesis$

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.

${\begin{matrix} 4 x + y > 6 \\ 3 x - y \leq 12 \end{matrix}$

ⓐ $open parenthesis 2, −1 close parenthesis$
ⓑ $open parenthesis 3, −2 close parenthesis$

ⓐ yes ⓑ yes

${\begin{matrix} y > \frac{1}{3} x + 2 \\ x - \frac{1}{4} y \leq 10 \end{matrix}$

ⓐ $open parenthesis 6, 5 close parenthesis$
ⓑ $open parenthesis 15, 8 close parenthesis$

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y \leq 3 x + 2 \\ y > - x - 1 \end{matrix}$

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.

${\begin{matrix} x - y > −1 \\ y < \frac{1}{3} x - 2 \end{matrix}$

${\begin{matrix} 2 x - 3 y < 6 \\ 3 x + 4 y \geq 12 \end{matrix}$

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.

${\begin{matrix} y \leq - \frac{3}{4} x + 1 \\ x \geq −5 \end{matrix}$

${\begin{matrix} x + 3 y < 5 \\ y \geq - \frac{1}{3} x + 6 \end{matrix}$

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.

${\begin{matrix} y \geq 2 x - 5 \\ −6 x + 3 y > −4 \end{matrix}$

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 26 bracelets and 14 necklaces?
ⓓ Should she display 39 bracelets and 1 necklace?

ⓐ ${\begin{matrix} b \geq 0 \\ n \geq 0 \\ b + n \leq 40 \\ 12 b + 18 n \geq 500 \end{matrix}$
ⓑ

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.

ⓒ yes
ⓓ 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 8 paperback books and 40 hardcover books?
ⓓ Can she buy 10 paperback books and 37 hardcover books?

Chapter Practice Test

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

${\begin{matrix} x - y = 5 \\ x + 2 y = −4 \end{matrix}$

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.

open parenthesis 2, −3 close parenthesis

${\begin{matrix} x - y > −2 \\ y \leq 3 x + 1 \end{matrix}$

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

${\begin{matrix} x + 4 y = 6 \\ −2 x + y = −3 \end{matrix}$

$open parenthesis 2, 1 close parenthesis$

${\begin{matrix} −3 x + 4 y = 25 \\ x - 5 y = −23 \end{matrix}$

${\begin{matrix} x + y - z = −1 \\ 2 x - y + 2 z = 8 \\ −3 x + 2 y + z = −9 \end{matrix}$

$open parenthesis 2, −2, 1 close parenthesis$

Solve the system of equations using a matrix.

${\begin{matrix} 2 x + y = 7 \\ x - 2 y = 6 \end{matrix}$

${\begin{matrix} −3 x + y + z = −4 \\ − x + 2 y - 2 z = 1 \\ 2 x - y - z = −1 \end{matrix}$

$open parenthesis 5, 7, 4 close parenthesis$

Solve using Cramer’s rule.

${\begin{matrix} 3 x + y = −3 \\ 2 x + 3 y = 6 \end{matrix}$

Evaluate the determinant by expanding by minors:
$| 3, −2, −2; 2, −1, 4; −1, 0, −3 |.$

$7$

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.
ⓒ Show the break-even point by graphing both the Revenue and Cost functions on the same grid.
ⓓ 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.
ⓒ Can she buy 20 candy bars and 40 lollipops?

ⓐ ${ C greater than or equal to 0; L greater than or equal to 0; C plus 0.5 L less than or equal to 50; L greater than or equal to 3 C$
ⓑ

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.

ⓒ no
ⓓ yes