Beginning and Intermediate Algebra

Julie Miller,Molly O’Neill,Nancy Hyde

Chapter 4

Systems of Linear Equations - all with Video Answers

Educators

Section 1

Solving Systems of Equations by the Graphing Method

Problem 1

Figure out your grade at this point. Are you earning the grade that you want? If not, maybe organizing a study group would help.
In a study group, check the activities that you might try to help you learn and understand the material.
___ Quiz each other by asking each other questions.
___Practice teaching each other.
___Share and compare class notes.
___Support and encourage each other.
___Work together on exercises and sample problems.

Christina M.

Christina M.

Numerade Educator

Problem 2

Define the key terms.
a. system of linear equations
b. solution to a system of linear equations
c. consistent system
d. inconsistent system
e. dependent system
f. independent system

Lauren S.

Numerade Educator

Problem 3

Determine if the given point is a solution to the system.
$$\begin{array}{ll}
3 x-y=7 & (2,-1) \\
x-2 y=4
\end{array}$$

Tristan W.

Numerade Educator

Problem 4

Determine if the given point is a solution to the system.
$$x-y=3\quad (4,1)$$
$$x+y=5$$

Tristan W.

Numerade Educator

Problem 5

Determine if the given point is a solution to the system.
$4 y=-3 x+12 \quad(0,4)$
$y=\frac{2}{3} x-4$

Tristan W.

Numerade Educator

Problem 6

Determine if the given point is a solution to the system.
$y=-\frac{1}{3} x+2 \quad(9,-1)$
$x=2 y+6$

Tristan W.

Numerade Educator

Problem 7

Determine if the given point is a solution to the system.
$\begin{aligned} 3 x-6 y &=9 \\ x-2 y &=3 \end{aligned} \quad\left(4, \frac{1}{2}\right)$

Tristan W.

Numerade Educator

Problem 8

Determine if the given point is a solution to the system.
$\begin{aligned} x-y &=4 \\ 3 x-3 y &=12 \end{aligned}$

Tristan W.

Numerade Educator

Problem 9

Determine if the given point is a solution to the system.
$\frac{1}{3} x=\frac{2}{5} y-\frac{4}{5} \quad(0,2)$
$\frac{3}{4} x+\frac{1}{2} y=2$

Tristan W.

Numerade Educator

Problem 10

Determine if the given point is a solution to the system.
$$\begin{aligned}
&\frac{1}{4} x+\frac{1}{2} y=\frac{3}{2} \quad(4,1)\\
&y=\frac{3}{2} x-6
\end{aligned}$$

Lauren S.

Numerade Educator

Problem 11

Match the graph of the system of equations with the appropriate description of the solution.
a. The solution set is $\{(1,3)\}$
b. { }
c. There are infinitely many solutions.
d. The solution set is $\{(0,0)\}$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 12

Match the graph of the system of equations with the appropriate description of the solution.
a. The solution set is $\{(1,3)\}$
b. { }
c. There are infinitely many solutions.
d. The solution set is $\{(0,0)\}$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 13

Match the graph of the system of equations with the appropriate description of the solution.
a. The solution set is $\{(1,3)\}$
b. { }
c. There are infinitely many solutions.
d. The solution set is $\{(0,0)\}$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 14

Match the graph of the system of equations with the appropriate description of the solution.
a. The solution set is $\{(1,3)\}$
b. { }
c. There are infinitely many solutions.
d. The solution set is $\{(0,0)\}$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 15

Graph each system of equations.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$

Lauren S.

Numerade Educator

Problem 16

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The lines are parallel.

Lauren S.

Numerade Educator

Problem 17

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The lines coincide.

Lauren S.

Numerade Educator

Problem 18

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The lines intersect at exactly one point.

Lauren S.

Numerade Educator

Problem 19

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The system is inconsistent.

Lauren S.

Numerade Educator

Problem 20

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The system is dependent.

Lauren S.

Numerade Educator

Problem 21

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The lines have the same slope but different $y$ -intercepts.

Lauren S.

Numerade Educator

Problem 22

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The lines have the same slope and same $y$ -intercept.

Lauren S.

Numerade Educator

Problem 23

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The lines have different slopes.

Lauren S.

Numerade Educator

Problem 24

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The system has exactly one solution.

Lauren S.

Numerade Educator

Problem 25

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The system has infinitely many solutions.

Lauren S.

Numerade Educator

Problem 26

Determine which system of equations (a, b, or c) makes the statement true.
a. $y=2 x-3$
$y=2 x+5$
b. $y=2 x+1$
$y=4 x-1$
c. $y=3 x-5$
$y=3 x-5$
The system has no solution.

Lauren S.

Numerade Educator

Problem 27

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&y=-x+4\\
&y=x-2
\end{aligned}$$

Tristan W.

Numerade Educator

Problem 28

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&y=3 x+2\\
&y=2 x
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 29

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&2 x+y=0\\
&3 x+y=1
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 30

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
x+y &=-1 \\
2 x-y &=-5
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 31

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
2 x+y &=6 \\
x &=1
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 32

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
4 x+3 y &=9 \\
x &=3
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 33

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{array}{r}
-6 x-3 y=0 \\
4 x+2 y=4
\end{array}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 34

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
2 x-6 y &=12 \\
-3 x+9 y &=12
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 35

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
-2 x+y &=3 \\
6 x-3 y &=-9
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 36

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
x+3 y &=0 \\
-2 x-6 y &=0
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 37

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&y=6\\
&2 x+3 y=12
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 38

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&y=-2\\
&x-2 y=10
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 39

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&x=4+y\\
&3 y=-3 x
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 40

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&3 y=4 x\\
&x-y=-1
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 41

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&-x+y=3\\
&4 y=4 x+6
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 42

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&x-y=4\\
&3 y=3 x+6
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 43

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&x=4\\
&2 y=4
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 44

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
-3 x &=6 \\
y &=2
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 45

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&2 x+y=4\\
&4 x-2 y=0
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 46

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&3 x+3 y=3\\
&2 x-y=5
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 47

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&y=0.5 x+2\\
&-x+2 y=4
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 48

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{array}{c}
3 x-4 y=6 \\
-6 x+8 y=-12
\end{array}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 49

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
&x-3 y=0\\
&y=-x-4
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 50

Solve the systems by graphing. If a system does not have a unique solution, identify the system as inconsistent or dependent.
$$\begin{aligned}
-6 x+3 y &=-6 \\
4 x+y &=-2
\end{aligned}$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 51

Costco Wholesale offers two types of memberships. The Executive Membership is $\$ 100$ a year, including an annual $2 \%$ reward. The Business Membership is $\$ 50$ a year without a reward. The total cost for membership, $y,$ depends on the amount of money spent on merchandise, $x$, and can be represented by the following equations:
Executive membership: $\quad y=100-0.02 x$ Business membership:
$y=50$
According to the graph, how much money spent on merchandise would result in the same cost for each membership?
(GRAPH CANNOT COPY)

Lauren S.

Numerade Educator

Problem 52

The cost to rent a $10 \mathrm{ft}$ by $10 \mathrm{ft}$ storage space is different for two different storage companics. The Storage Bin charges $\$ 90$ per month plus a nonrefundable deposit of $\$ 120$. AAA Storage charges $\$ 110$ per month with no deposit. The total cost, $y,$ to rent a $10 \mathrm{ft}$ by $10 \mathrm{ft}$ space depends on the number of months, $x,$ according to the equations The Storage Bin: $\quad y=90 x+120$ AAA Storage: $\quad y=110 x$
From the graph, determine the number of months required for which the cost to rent space is equal for both companies.
(GRAPH CANNOT COPY)

Lauren S.

Numerade Educator

Problem 53

Explain why the ordered pair cannot be a solution to the system of equations.
$$(-3,1)$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 54

Explain why the ordered pair cannot be a solution to the system of equations.
$$(-1,-4)$$
(GRAPH CANNOT COPY)

Tristan W.

Numerade Educator

Problem 55

Write a system of linear equations whose solution set is $\{(2,1)\}$.

Lauren S.

Numerade Educator

Problem 56

Write a system of linear equations whose solution set is $[(1,4)]$.

Tristan W.

Numerade Educator

Problem 57

One equation in a system of linear equations is $x+y=4$. Write a second equation such that the system will have no solution. (Answers may vary.)

Tristan W.

Numerade Educator

Problem 58

One equation in a system of linear equations is $x-y=3 .$ Write a second equation such that the system will have infinitely many solutions. (Answers may vary.)

Tristan W.

Numerade Educator