Chapter 7

Systems of Equations and Inequalities

Educators

MB
SM

Problem 1

Can a system of linear equations have exactly two solutions? Explain why or why not.

Tony W.
Numerade Educator

Problem 2

If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company's profit margins.

MB
Matt B.
Numerade Educator

Problem 3

If you are solving a break-even analysis and get a negative break-even point, explain what this
signifies for the company?

Tony W.
Numerade Educator

Problem 4

If you are solving a break-even analysis and there is no break-even point, explain what this means
for the company. How should they ensure there is a break-even point?

MB
Matt B.
Numerade Educator

Problem 5

Given a system of equations, explain at least two different methods of solving that system.

Tony W.
Numerade Educator

Problem 6

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.
$$
\begin{array}{l}{5 x-y=4} \\ {x+6 y=2 \text { and }(4,0)}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 7

For the following exercises, determine whether the given ordered pair is a solution to the system of equations
$$
\begin{array}{l}{-3 x-5 y=13} \\ {-x+4 y=10 \text { and }(-6,1)}\end{array}
$$

Tony W.
Numerade Educator

Problem 8

For the following exercises, determine whether the given ordered pair is a solution to the system of equations
$$
\begin{array}{l}{3 x+7 y=1} \\ {2 x+4 y=0 \text { and }(2,3)}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 9

For the following exercises, determine whether the given ordered pair is a solution to the system of equations
$$
\begin{aligned}-2 x+5 y &=7 \\ 2 x+9 y &=7 \text { and }(-1,1) \end{aligned}
$$

Tony W.
Numerade Educator

Problem 10

For the following exercises, determine whether the given ordered pair is a solution to the system of equations
$$
\begin{aligned} x+8 y &=43 \\ 3 x-2 y &=-1 \text { and }(3,5) \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 11

For the following exercises, solve each system by substitution.
$$
\begin{aligned} x+3 y &=5 \\ 2 x+3 y &=4 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 12

For the following exercises, solve each system by substitution.
$$
\begin{array}{c}{3 x-2 y=18} \\ {5 x+10 y=-10}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 13

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{4 x+2 y=-10} \\ {3 x+9 y=0}\end{array}
$$

Tony W.
Numerade Educator

Problem 14

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{2 x+4 y=-3.8} \\ {9 x-5 y=1.3}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 15

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{-2 x+3 y=1.2} \\ {-3 x-6 y=1.8}\end{array}
$$

Tony W.
Numerade Educator

Problem 16

For the following exercises, solve each system by substitution.
$$
\begin{aligned} x-0.2 y &=1 \\-10 x+2 y &=5 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 17

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{3 \quad x+5 y=9} \\ {30 x+50 y=-90}\end{array}
$$

Tony W.
Numerade Educator

Problem 18

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{-3 x+y=2} \\ {12 x-4 y=-8}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 19

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{\frac{1}{2} x+\frac{1}{3} y=16} \\ {\frac{1}{6} x+\frac{1}{4} y=9}\end{array}
$$

Tony W.
Numerade Educator

Problem 20

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{-\frac{1}{4} x+\frac{3}{2} y=11} \\ {-\frac{1}{8} x+\frac{1}{3} y=3}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 21

For the following exercises, solve each system by addition.
$$
\begin{aligned}-2 x+5 y &=-42 \\ 7 x+2 y &=30 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 22

For the following exercises, solve each system by addition.
$$
\begin{array}{l}{6 x-5 y=-34} \\ {2 x+6 y=4}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 23

For the following exercises, solve each system by addition.
$$
\begin{aligned} 5 x-y &=-2.6 \\-4 x-6 y &=1.4 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 24

For the following exercises, solve each system by addition.
$$
\begin{array}{l}{7 x-2 y=3} \\ {4 x+5 y=3.25}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 25

For the following exercises, solve each system by addition.
$$
\begin{array}{l}{-x+2 y=-1} \\ {5 x-10 y=6}\end{array}
$$

Tony W.
Numerade Educator

Problem 26

For the following exercises, solve each system by addition.
$$
\begin{aligned} 7 x+6 y &=2 \\-28 x-24 y &=-8 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 27

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{\frac{5}{6} x+\frac{1}{4} y=0} \\ {\frac{1}{8} x-\frac{1}{2} y=-\frac{43}{120}}\end{array}
$$

Tony W.
Numerade Educator

Problem 28

For the following exercises, solve each system by addition.
$$
\begin{aligned} \frac{1}{3} x+\frac{1}{9} y &=\frac{2}{9} \\-\frac{1}{2} x+\frac{4}{5} y &=-\frac{1}{3} \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 29

For the following exercises, solve each system by addition.
$$
\begin{aligned}-0.2 x+0.4 y &=0.6 \\ x-2 y &=-3 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 30

For the following exercises, solve each system by addition.
$$
\begin{aligned}-0.1 x+0.2 y &=0.6 \\ 5 x-10 y &=1 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 31

For the following exercises, solve each system by any method.
$$
\begin{aligned} 5 x+9 y &=16 \\ x+2 y &=4 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 32

For the following exercises, solve each system by any method.
$$
\begin{array}{l}{6 x-8 y=-0.6} \\ {3 x+2 y=0.9}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 33

For the following exercises, solve each system by any method.
$$
\begin{array}{l}{5 x-2 y=2.25} \\ {7 x-4 y=3}\end{array}
$$

Tony W.
Numerade Educator

Problem 34

For the following exercises, solve each system by any method.
$$
\begin{aligned} x-\frac{5}{12} y &=-\frac{55}{12} \\-6 x+\frac{5}{2} y &=\frac{55}{2} \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 35

For the following exercises, solve each system by any method.
$$
\begin{array}{l}{7 x-4 y=\frac{7}{6}} \\ {2 x+4 y=\frac{1}{3}}\end{array}
$$

Tony W.
Numerade Educator

Problem 36

For the following exercises, solve each system by any method.
$$
\begin{array}{l}{3 x+6 y=11} \\ {2 x+4 y=9}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 37

For the following exercises, solve each system by any method.
$$
\begin{aligned} \frac{7}{3} x-\frac{1}{6} y &=2 \\-\frac{21}{6} x+\frac{3}{12} y &=-3 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 38

For the following exercises, solve each system by any method.
$$
\begin{array}{l}{\frac{1}{2} x+\frac{1}{3} y=\frac{1}{3}} \\ {\frac{3}{2} x+\frac{1}{4} y=-\frac{1}{8}}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 39

For the following exercises, solve each system by any method.
$$
\begin{array}{l}{2.2 x+1.3 y=-0.1} \\ {4.2 x+4.2 y=2.1}\end{array}
$$

Tony W.
Numerade Educator

Problem 40

For the following exercises, solve each system by any method.
$$
\begin{aligned} 0.1 x+0.2 y &=2 \\ 0.35 x-0.3 y &=0 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 41

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.
$$
\begin{array}{l}{3 x-y=0.6} \\ {x-2 y=1.3}\end{array}
$$

Tony W.
Numerade Educator

Problem 42

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.
$$
\begin{array}{l}{-x+2 y=4} \\ {2 x-4 y=1}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 43

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.
$$
\begin{aligned} x+2 y &=7 \\ 2 x+6 y &=12 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 44

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.
$$
\begin{aligned} 3 x-5 y &=7 \\ x-2 y &=3 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 45

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.
$$
\begin{aligned} 3 x-2 y &=5 \\-9 x+6 y &=-15 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 46

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.
$$
\begin{aligned} 0.1 x+0.2 y &=0.3 \\-0.3 x+0.5 y &=1 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 47

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.
$$
\begin{aligned}-0.01 x+0.12 y &=0.62 \\ 0.15 x+0.20 y &=0.52 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 48

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.
$$
\begin{aligned} 0.5 x+0.3 y &=4 \\ 0.25 x-0.9 y &=0.46 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 49

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.
$$
\begin{aligned} 0.15 x+0.27 y &=0.39 \\-0.34 x+0.56 y &=1.8 \end{aligned}
$$

Tony W.
Numerade Educator

Problem 50

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.
$$
\begin{aligned}-0.71 x+0.92 y &=0.13 \\ 0.83 x+0.05 y &=2.1 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 51

For the following exercises, solve each system in terms of $A, B, C, D, E,$ and $F$ where $A-F$ are nonzero numbers. Note that $A \neq B$ and $A E \neq B D .$
$$
\begin{array}{l}{x+y=A} \\ {x-y=B}\end{array}
$$

Tony W.
Numerade Educator

Problem 52

For the following exercises, solve each system in terms of $A, B, C, D, E,$ and $F$ where $A-F$ are nonzero numbers. Note that $A \neq B$ and $A E \neq B D .$
$$
\begin{array}{l}{x+A y=1} \\ {x+B y=1}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 53

For the following exercises, solve each system in terms of $A, B, C, D, E,$ and $F$ where $A-F$ are nonzero numbers. Note that $A \neq B$ and $A E \neq B D .$
$$
\begin{array}{l}{A x+y=0} \\ {B x+y=1}\end{array}
$$

Tony W.
Numerade Educator

Problem 54

For the following exercises, solve each system in terms of $A, B, C, D, E,$ and $F$ where $A-F$ are nonzero numbers. Note that $A \neq B$ and $A E \neq B D .$
$$
\begin{array}{r}{A x+B y=C} \\ {x+y=1}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 55

For the following exercises, solve each system in terms of $A, B, C, D, E,$ and $F$ where $A-F$ are nonzero numbers. Note that $A \neq B$ and $A E \neq B D .$
$$
\begin{array}{l}{A x+B y=C} \\ {D x+E y=F}\end{array}
$$

Tony W.
Numerade Educator

Problem 56

For the following exercises, solve for the desired quantity.
A stuffed animal business has a total cost of production $C=12 x+30$ and a revenue function $R=20 x .$ Find the break-even point.

MB
Matt B.
Numerade Educator

Problem 57

For the following exercises, solve for the desired quantity.
A fast-food restaurant has a cost of production $C(x)=11 x+120$ and a revenue function $R(x)=5 x$ . When does the company start to turn a profit?

Tony W.
Numerade Educator

Problem 58

For the following exercises, solve for the desired quantity.
A cell phone factory has a cost of production $C(x)=150 x+10,000$ and a revenue function $R(x)=200 x .$ What is the break-even point?

MB
Matt B.
Numerade Educator

Problem 59

For the following exercises, solve for the desired quantity.
A musician charges $C(x)=64 x+20,000$ , where $x$ is the total number of attendees at the concert. The venue charges $\$ 80$ per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

SM
Suguna M.
Numerade Educator

Problem 60

For the following exercises, solve for the desired quantity.
A guitar factory has a cost of production $C(x)=75 x+50,000$ . If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

MB
Matt B.
Numerade Educator

Problem 61

For the following exercises, use a system of linear equations with two variables and two equations to solve.
Find two numbers whose sum is 28 and difference is $13 .$

Tony W.
Numerade Educator

Problem 62

For the following exercises, use a system of linear equations with two variables and two equations to solve.
A number is 9 more than another number. Twice the sum of the two numbers is $10 .$ Find the two
numbers.

MB
Matt B.
Numerade Educator

Problem 63

For the following exercises, use a system of linear equations with two variables and two equations to solve.
The startup cost for a restaurant is $\$ 120,000$ , and each meal costs $\$ 10$ for the restaurant to make. If each meal is then sold for $\$ 15$ , after how many meals does the restaurant break even?

Tony W.
Numerade Educator

Problem 64

For the following exercises, use a system of linear equations with two variables and two equations to solve.
A moving company charges a flat rate of $\$ 150,$ and an additional $\$ 5$ for each box. If a taxi service would charge $\$ 20$ for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

MB
Matt B.
Numerade Educator

Problem 65

For the following exercises, use a system of linear equations with two variables and two equations to solve.
A total of $1,595$ first- and second-year college students gathered at a pep rally. The number of freshmen exceeded the number of sophomores by $15 .$ How many freshmen and sophomores were in
attendance?

Tony W.
Numerade Educator

Problem 66

For the following exercises, use a system of linear equations with two variables and two equations to solve.
276 students enrolled in a freshman-level chemistry class. By the end of the semester, 5 times the number of students passed as failed. Find the number of students who passed, and the number of students who failed.

MB
Matt B.
Numerade Educator

Problem 67

For the following exercises, use a system of linear equations with two variables and two equations to solve.
There were 130 faculty at a conference. If there were 18 more women than men attending, how many of
each gender attended the conference?

Tony W.
Numerade Educator

Problem 68

For the following exercises, use a system of linear equations with two variables and two equations to solve.
A jeep and BMW enter a highway running east- west at the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 7 mph slower than the BMW. After 2 hours from the time the BMW entered the highway, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.

MB
Matt B.
Numerade Educator

Problem 69

For the following exercises, use a system of linear equations with two variables and two equations to solve.
If a scientist mixed 10$\%$ saline solution with 60 saline solution to get 25 gallons of 40$\%$ saline solution, how many gallons of 10$\%$ and 60$\%$ solutions were mixed?

Tony W.
Numerade Educator

Problem 70

For the following exercises, use a system of linear equations with two variables and two equations to solve.
An investor earned triple the profits of what she earned last year. If she made $\$ 500,000.48$ total for both years, how much did she earn in profits each year?

MB
Matt B.
Numerade Educator

Problem 71

For the following exercises, use a system of linear equations with two variables and two equations to solve.
An investor who dabbles in real estate invested 1.1 million dollars into two land investments. On the first investment, Swan Peak, her return was a 110$\%$ increase on the money she invested. On the second investment, Riverside Community, she earned 50$\%$ over what she invested. If she earned $\$ 1$ million in profits, how much did she invest in each of the land deals?

Tony W.
Numerade Educator

Problem 72

For the following exercises, use a system of linear equations with two variables and two equations to solve.
If an investor invests a total of $\$ 25,000$ into two bonds, one that pays 3$\%$ simple interest, and the other that pays 2$\frac{7}{8} \%$ interest, and the investor earns $\$ 737.50$ annual interest, how much was invested in each account?

MB
Matt B.
Numerade Educator

Problem 73

For the following exercises, use a system of linear equations with two variables and two equations to solve.
If an investor invests $\$ 23,000$ into two bonds, one that pays 4$\%$ in simple interest, and the other paying 2$\%$ simple interest, and the investor earns $\$ 710.00$ annual interest, how much was invested in each account?

Tony W.
Numerade Educator

Problem 74

For the following exercises, use a system of linear equations with two variables and two equations to solve.
CDs cost 55.96 more than DVDs at All Bets Are Off Electronics. How much would 6 $\mathrm{CDs}$ and 2 $\mathrm{DVDs}$ cost if 5 $\mathrm{CD}$ s and 2 $\mathrm{DVDs} \operatorname{cost} \$ 127.73 ?$

MB
Matt B.
Numerade Educator

Problem 75

For the following exercises, use a system of linear equations with two variables and two equations to solve.
A store clerk sold 60 pairs of sneakers. The high-tops sold for $\$ 98.99$ and the low-tops sold for $\$ 129.99 .$ If the receipts for the two types of sales totaled $\$ 6,404.40$ , how many of each type of sneaker were sold?

Tony W.
Numerade Educator

Problem 76

For the following exercises, use a system of linear equations with two variables and two equations to solve.
A concert manager counted 350 ticket receipts the day after a concert. The price for a student ticket was $\$ 12.50$ , and the price for an adult ticket was $\$ 16.00$ . The register confirms that $\$ 5,075$ was taken in. How many student tickets and adult tickets were sold?

MB
Matt B.
Numerade Educator

Problem 77

For the following exercises, use a system of linear equations with two variables and two equations to solve.
Admission into an amusement park for 4 children and 2 adults is $\$ 116.90 .$ For 6 children and 3 adults, the admission is $\$ 175.35$ . Assuming a different price for children and adults, what is the price of the child's ticket and the price of the adult ticket?

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