Algebra 2

Prentice Hall, Basia Kennedy, Dan Ramirez

Chapter 3

Linear Systems

Educators

KB
AG

Problem 1

Solve each system by graphing. Check your answers.
$$
\left\{\begin{array}{l}{y=x-2} \\ {y=-2 x+7}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 2

Solve each system by graphing. Check your answers.
$$
\left\{\begin{array}{l}{y=-x+3} \\ {y=\frac{3}{2} x-2}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 3

Solve each system by graphing. Check your answers.
$$
\left\{\begin{aligned} 2 x+4 y &=12 \\ x+y &=2 \end{aligned}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 4

Solve each system by graphing. Check your answers.
$$
\left\{\begin{array}{l}{x=-3} \\ {y=5}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 5

Solve each system by graphing. Check your answers.
$$
\left\{\begin{array}{l}{2 x-2 y=4} \\ {y-x=6}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 6

Solve each system by graphing. Check your answers.
$$
\left\{\begin{array}{c}{3 x+y=5} \\ {x-y=7}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 7

Solve each system by graphing. Check your answers.
$$
\left\{\begin{aligned}-5 x+y &=-9 \\ x+3 y &=21 \end{aligned}\right.
$$

KB
Karan B.
Numerade Educator

Problem 8

Solve each system by graphing. Check your answers.
$$
\left\{\begin{array}{l}{y=x} \\ {y-5 x=0}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 9

Solve each system by graphing. Check your answers.
$$
\left\{\begin{array}{l}{x=10} \\ {x=y-10}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 10

For Exercises $10-11,$ use your graphing calculator. Find linear models for each set of data. Use each model to predict the year in which the quantities will be equal.
Annual U.S. Consumption of Vegetables
(PICTURE NOT COPY)

Weston B.
Numerade Educator

Problem 11

For Exercises $10-11,$ use your graphing calculator. Find linear models for each set of data. Use each model to predict the year in which the quantities will be equal.
U.S. Life Expectancy at Birth
(PICTURE NOT COPY)

KB
Karan B.
Numerade Educator

Problem 12

a. Business The spreadsheet shows the monthly revenue and monthly expenses for a new business. Find a linear model for monthly revenue and a linear model for monthly expenses.
b. Use the models to predict the month in which revenue will equal expenses.

Weston B.
Numerade Educator

Problem 13

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{aligned} 7 x-y &=6 \\-7 x+y &=-6 \end{aligned}\right.
$$

KB
Karan B.
Numerade Educator

Problem 14

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{aligned}-3 x+y &=4 \\ x-\frac{1}{3} y &=1 \end{aligned}\right.
$$

Weston B.
Numerade Educator

Problem 15

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{aligned} 4 x+8 y &=12 \\ x+2 y &=-3 \end{aligned}\right.
$$

KB
Karan B.
Numerade Educator

Problem 16

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{array}{l}{y=2 x-1} \\ {y=-2 x+5}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 17

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{array}{l}{x=6} \\ {x=-2}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 18

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{array}{l}{2 y=5 x+6} \\ {-10 x+4 y=8}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 19

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{aligned} x-3 y &=2 \\ 4 x-12 y &=8 \end{aligned}\right.
$$

KB
Karan B.
Numerade Educator

Problem 20

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{array}{l}{x+4 y=12} \\ {2 x-8 y=4}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 21

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{array}{l}{4 x+8 y=-6} \\ {6 x+12 y=-9}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 22

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{array}{l}{4 y-2 x=6} \\ {8 y=4 x-12}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 23

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{array}{l}{y-x=0} \\ {y=-x}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 24

Without graphing, classify each system as independent, dependent, or inconsistent.
$$
\left\{\begin{array}{l}{2 y-x=4} \\ {\frac{1}{2} x-y=2}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 25

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{3=4 y+x} \\ {4 y=-x+3}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 26

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{x-2 y+1=0} \\ {x+4 y-6=0}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 27

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{3 x+6 y-12=0} \\ {x+2 y=8}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 28

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{aligned}-x+3 y &=6 \\ 2 x-y &=8 \end{aligned}\right.
$$

Weston B.
Numerade Educator

Problem 29

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{3 x+y=3} \\ {2 x-y=7}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 30

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{2 x+3 y=6} \\ {4 x=6 y+3}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 31

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{10-3 x=-3 y} \\ {2=2 x+y}\end{array}\right.
$$

AG
Ankit G.
Numerade Educator

Problem 32

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{3 x=-5 y+4} \\ {250+150 x=300}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 33

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{aligned} x+3 y &=6 \\ 6 y+2 x &=12 \end{aligned}\right.
$$

KB
Karan B.
Numerade Educator

Problem 34

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{2 y+x=8} \\ {y-2 x=-6}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 35

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{y=-2 x+6} \\ {x-3 y=-6}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 36

Graph and solve each system. Where necessary, estimate the solution.
$$
\left\{\begin{array}{l}{-x-2=-2 y} \\ {2 x-4 y-4=0}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 37

Banking To pay your monthly bills, you can either open a checking account or use an online banking service. A local bank charges $\$ 3$ per month and $\$ .40$ per check, while an online services charges a flat fee of $\$ 9$ per month.
a. Write and graph a system of linear equations to model the cost $c$ of each service for $b$ bills that you need to pay monthly.
b. Find the point of intersection of the two linear models. What does this answer represent?
c. If you pay about 12 bills per month, which service should you choose? Explain.

KB
Karan B.
Numerade Educator

Problem 38

Classify each system without graphing.
$$
\left\{\begin{array}{l}{3 x-2 y=8} \\ {4 y=6 x-5}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 39

Classify each system without graphing.
$$
\left\{\begin{array}{l}{2 x+8 y=6} \\ {x=-4 y+3}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 40

Classify each system without graphing.
$$
\left\{\begin{array}{l}{3 a+6 b=14} \\ {-a+2 b=3}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 41

Classify each system without graphing.
$$
\left\{\begin{array}{l}{3 m=-5 n+4} \\ {n-\frac{6}{5}=-\frac{3}{5} m}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 42

Classify each system without graphing.
$$
\left\{\begin{array}{l}{-12 x+4 y=8} \\ {y-4=3 x}\end{array}\right.
$$

Weston B.
Numerade Educator

Problem 43

Classify each system without graphing.
$$
\left\{\begin{array}{l}{-6 y+18=12 x} \\ {3 y+6 x=9}\end{array}\right.
$$

KB
Karan B.
Numerade Educator

Problem 44

Fees Suppose you are going on vacation and leaving your dog in a kennel.The Bowowery charges $\$ 5$ per day, which includes a one-time grooming treatment. The Poochpad charges $\$ 20$ per day and a one-time fee of $\$ 30$ for grooming.
a. Write a system of equations to represent the cost $c$ for $d$ days that your dog will stay at a kennel.
b. Using a graphing calculator, find the number of days for which the costs are the same.
c. If your vacation is a week long, which kennel should you choose? Explain.

Weston B.
Numerade Educator

Problem 45

Advertising You and your business partner are mailing advertising flyers to your customers. You address 6 flyers each minute and have already done $80 .$ Your partner addresses 4 flyers each minute and has already done $100 .$ Graph and solve a system of equations to find when the two of you will have
addressed equal numbers of flyers.

KB
Karan B.
Numerade Educator

Problem 46

Open-Ended Write a second equation for each system so that the system will have the indicated number of solutions.
$$
\begin{array}{l}{\text { one }} \\ {\left\{\begin{array}{l}{y=-3 x+2} \\ {?}\end{array}\right.}\end{array}
$$

Weston B.
Numerade Educator

Problem 47

Open-Ended Write a second equation for each system so that the system will have the indicated number of solutions.
$$
\begin{array}{l}{\text { none }} \\ {\left\{\begin{array}{c}{y=-4 x-6} \\ {?}\end{array}\right.}\end{array}
$$

KB
Karan B.
Numerade Educator

Problem 48

Open-Ended Write a second equation for each system so that the system will have the indicated number of solutions.
$$
\begin{array}{l}{\text { an infinite number }} \\ {\left\{\begin{array}{c}{3 y=6 x+7} \\ {?}\end{array}\right.}\end{array}
$$

Weston B.
Numerade Educator

Problem 49

Reasoning Is it possible for an inconsistent linear system to consist of two lines with the same $y$ -intercept? Explain.

KB
Karan B.
Numerade Educator

Problem 50

Writing Summarize the possible relationships for the $y$ -intercepts, slopes, and number of solutions in a system of two linear equations of two variables.

Weston B.
Numerade Educator

Problem 51

Open-Ended Write a second equation for each system so that the system will have the indicated number of solutions.
$$
\begin{array}{l}{\text { infinite number of solutions }} \\ {\left\{\frac{x}{4}+\frac{y}{3}=1\right.} \\ {\text { ? }}\end{array}
$$

KB
Karan B.
Numerade Educator

Problem 52

Open-Ended Write a second equation for each system so that the system will have the indicated number of solutions.
$$
\begin{array}{l}{\text { no solutions }} \\ {\left\{\begin{array}{c}{5 x+2 y=10} \\ {?}\end{array}\right.}\end{array}
$$

Weston B.
Numerade Educator

Problem 53

Write a system of linear equations with the solution set $\{(x, y) | y=5 x+2\}$

KB
Karan B.
Numerade Educator

Problem 54

Critical Thinking Look back through the exercises on the previous two pages to find several dependent systems. What relationship exists between the equations in each system?

Weston B.
Numerade Educator

Problem 55

Economics Resarch shows that in a certain market only 2000 widgets can be sold at $\$ 8$ each, but if the price is reduced to $\$ 3,$ then $10,000$ can be sold.
a. Let $p$ represent price and $n$ represent the number of widgets. Identify the independent variable and the dependent variable.
b. Use the information above to write a linear demand equation.
c. A shop can make 2000 widgets for $\$ 5$ each and $20,000$ widgets for $\$ 2$ each. Use this information to write a linear supply equation.
d. Find the equilibrium point where supply is equal to demand and profit is a maximum. Explain the meaning of the coordinates of this point within the context of the exercise.

KB
Karan B.
Numerade Educator

Problem 56

Which is an equation for Line 2$?$
$$\begin{array}{ll}{\text { A. } 3 x-5 y=15} & {\text { B. } 3 x+5 y=3} \\ {\text { C. } 3 x+5 y=15} & {\text { D. } 5 x+3 y=15}\end{array}$$

Weston B.
Numerade Educator

Problem 57

Which is NOT an equation for Line 1$?$
$$\begin{array}{ll}{\text { F. } y=x} & {\text { G. } x+y=0} \\ {\text { H. } x-y=0} & {\text { J. } y-x=0}\end{array}$$

KB
Karan B.
Numerade Educator

Problem 58

Which point lies on both Line 1 and Line 2$?$
$$\begin{array}{ll}{\text { A. }(0,0)} & {\text { B. }(1.875,1.875)} \\ {\text { C. }(1.95,1.95)} & {\text { D. }(2,2)}\end{array}$$

Weston B.
Numerade Educator

Problem 59

What is the solution of the system? $\left\{\begin{array}{c}{5 x+6 y=-24} \\ {-2 x+3 y=15} \\ {\text { H. }(-6,1)}\end{array}\right.$
$$
\begin{array}{llll}{\text { F. }(6,-1)} & {\text { G. }(6,1)} & {\text { H. }(-6,1)} & {\text { J. }(-6,-1)}\end{array}
$$

KB
Karan B.
Numerade Educator

Problem 60

Explain how you can use slopes to show that the system $\left\{\begin{array}{l}{2 x-5 y=23} \\ {3 y-7 x=-8}\end{array}\right.$ is NOT inconsistent.

Weston B.
Numerade Educator

Problem 61

One equation of a system of equations is $2 x-3 y=5$
a. Find a second equation such that the system is dependent.
b. Find a second equation such that the system is inconsistent.

KB
Karan B.
Numerade Educator

Problem 62

Graph each inequality on a coordinate plane.
$$
3 x-4 y \geq 16
$$

Weston B.
Numerade Educator

Problem 63

Graph each inequality on a coordinate plane.
$$
-5 x>8 y+4
$$

KB
Karan B.
Numerade Educator

Problem 64

Graph each inequality on a coordinate plane.
$$
x<-4
$$

Weston B.
Numerade Educator

Problem 65

Write an equation for each line.
$$
m=-\frac{2}{3} ; \text { contains }(-9,4)
$$

KB
Karan B.
Numerade Educator

Problem 66

Write an equation for each line.
$$
m=0 ; \text { contains }(3,4)
$$

Weston B.
Numerade Educator

Problem 67

Write an equation for each line.
$$
m=2 ; \text { contains }(-2,-3)
$$

KB
Karan B.
Numerade Educator

Problem 68

Write an equation for each line.
$$
m=-\frac{1}{2} ; \text { contains }(2,-6)
$$

Weston B.
Numerade Educator

Problem 69

Solve each equation and check the solution.
$$
3 n=-4(2+n)
$$

KB
Karan B.
Numerade Educator

Problem 70

Solve each equation and check the solution.
$$
-4 a+a=7 a-6
$$

Weston B.
Numerade Educator

Problem 71

Solve each equation and check the solution.
$$
\frac{x}{3}+5=\frac{1}{6}
$$

KB
Karan B.
Numerade Educator

Problem 72

Solve each equation and check the solution.
$$
4 x-2=\frac{1}{2} x
$$

Weston B.
Numerade Educator

Problem 73

Solve each equation and check the solution.
$$
\frac{r}{5}+5=r-3
$$

AG
Ankit G.
Numerade Educator

Problem 74

Solve each equation and check the solution.
$$
2(m-3)=-4
$$

Weston B.
Numerade Educator