Elementary and Intermediate Algebra 5th

Alan S. Tussy, R. David Gustafson

Chapter 4

Systems of Linear Equations and Inequalities - all with Video Answers

Educators

IM

Section 1

Solving Systems of Equations by Graphing

Problem 1

Fill in the blanks.
The pair of equations $\left\{\begin{array}{l}{x-y=-1} \\ {2 x-y=1}\end{array}\right.$ is called a ___ linear equations.

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Problem 2

Fill in the blanks.
Because the ordered pair $(2,3)$ satisfics both equations in Problem 1, it is called a ___.

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Problem 3

Fill in the blanks.
The point of __of the lines graphed in part (a) below
is $(1,2)$
(GRAPH CANNOT COPY)

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Problem 4

The lines graphed in part (b) above do not intersect. They are ___lines.

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Problem 5

Fill in the blanks.
A system of equations that has at least one solution is called a
___ system. A ___system with no solution is called an system.

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Problem 6

Fill in the blanks.
We call equations with different graphs___. Because ___ $=$ equation, they have the same graph..

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Problem 7

Fill in the blanks.
Refer to the illustration below.
a. If the coordinates of point $A$ are substituted into the equation for Line $1,$ will the result be true or false?
b. If the coordinates of point $C$ are substituted into the equation for Line $1,$ will the result be true or false?
(ILLUSTRATION CANNOT COPY)

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Problem 8

Fill in the blanks.
Refer to the illustration below.
a. If the coordinates of point $C$ are substituted into the equation for Line $2,$ will the result be true or false?
b. If the coordinates of point $B$ are substituted into the equation for Line
1, will the result be true or false?
(ILLUSTRATION CANNOT COPY)

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Problem 9

a. To graph $5 x-2 y=10,$ we can use the intercept method. Complete the table.
1, will the result be true or false?(TABLE CANNOT COPY)
b. To graph $y=3 x-2,$ we can use the slope and $y$ -intercept. Fill in the blanks.

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Problem 10

What is the apparent solution of the system graphed in figure
(a) below? Is the system consistent or inconsistent?
(FIGURE CANNOT COPY)

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Problem 11

How many solutions does the system graphed in figure (b) above have? Are the equations dependent or independent?

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Problem 12

How many solutions does the system graphed on the right have? Give three of the solutions. Is the system consistent or inconsistent?
(GRAPH CANNOT COPY)

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Problem 13

Determine whether the ordered pair is a solution of the given system of equations.
$$
(1,1),\left\{\begin{array}{l}
{x+y=2} \\
{2 x-y=1}
\end{array}\right.
$$

IM

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Problem 14

Determine whether the ordered pair is a solution of the given system of equations.
$$
(1,3),\left\{\begin{array}{l}
{2 x+y=5} \\
{3 x-y=0}
\end{array}\right.
$$

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Problem 15

Determine whether the ordered pair is a solution of the given system of equations.
$$
(3,-2),\left\{\begin{array}{l}
{2 x+y=4} \\
{y=1-x}
\end{array}\right.
$$

IM

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Problem 16

Determine whether the ordered pair is a solution of the given system of equations.
$$
(-2,4),\left\{\begin{array}{l}
{2 x+2 y=4} \\
{3 y=10-x}
\end{array}\right.
$$

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Problem 17

Determine whether the ordered pair is a solution of the given system of equations.
$$
(12,0),\left\{\begin{array}{l}
{x-9 y=12} \\
{y=10-x}
\end{array}\right.
$$

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Problem 18

$$
(15,0),\left\{\begin{array}{l}
{x-2 y=15} \\
{y=16-x}
\end{array}\right.
$$

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Problem 19

Determine whether the ordered pair is a solution of the given system of equations.
$$
(-2,-4),\left\{\begin{array}{l}
{4 x+5 y=-23} \\
{-3 x+2 y=0}
\end{array}\right.
$$

IM

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Problem 20

Determine whether the ordered pair is a solution of the given system of equations.
$$
(-5,2),\left\{\begin{array}{l}
{-2 x+7 y=17} \\
{3 x-4 y=-19}
\end{array}\right.
$$

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Problem 21

Determine whether the ordered pair is a solution of the given system of equations.
$$
\left(\frac{1}{2}, 3\right),\left\{\begin{array}{l}
{2 x+y=4} \\
{4 x-11=3 y}
\end{array}\right.
$$

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Problem 22

$$
\left(2, \frac{1}{3}\right),\left\{\begin{array}{l}
{x-3 y=1} \\
{-2 x+6=-6 y}
\end{array}\right.
$$

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Problem 23

Determine whether the ordered pair is a solution of the given system of equations.
$$
(2.5,3.5),\left\{\begin{array}{l}
{4 x-3=2 y} \\
{4 y+1=6 x}
\end{array}\right.
$$

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Problem 24

Determine whether the ordered pair is a solution of the given system of equations.
$$
(0.2,0.3),\left\{\begin{array}{l}
{20 x+10 y=7} \\
{20 y=15 x+3}
\end{array}\right.
$$

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Problem 25

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{2 x+3 y=12} \\
{2 x-y=4}
\end{array}\right.
$$

IM

Ian M.

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Problem 26

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{5 x+y=5} \\
{5 x+3 y=15}
\end{array}\right.
$$

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Problem 27

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x+y=4} \\
{x-y=-6}
\end{array}\right.
$$

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Problem 28

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x+y=4} \\
{x-y=-2}
\end{array}\right.
$$

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Problem 29

Solve each system by graphing. If a system has no solution or infinitely many solutions, so state.
$$
\left\{\begin{array}{l}
{y=-\frac{1}{3} x-4} \\
{x+3 y=6}
\end{array}\right.
$$

IM

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Numerade Educator

Problem 30

Solve each system by graphing. If a system has no solution or infinitely many solutions, so state.
$$
\left\{\begin{array}{l}
{y=-\frac{1}{2} x-3} \\
{x+2 y=2}
\end{array}\right.
$$

IM

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Problem 31

Solve each system by graphing. If a system has no solution or infinitely many solutions, so state.
$$
\left\{\begin{array}{l}
{y=3 x} \\
{y-3 x=-3}
\end{array}\right.
$$

IM

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Problem 32

Solve each system by graphing. If a system has no solution or infinitely many solutions, so state.
$$
\left\{\begin{array}{l}
{y=-2 x} \\
{2 x+y=-2}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 33

Solve each system by graphing. If a system has no solution or infinitely many solutions, so state.
$$
\left\{\begin{array}{l}
{y=x-1} \\
{3 x-3 y=3}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 34

Solve each system by graphing. If a system has no solution or infinitely many solutions, so state.
$$
\left\{\begin{array}{l}
{y=-x+1} \\
{4 x+4 y=4}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 35

Solve each system by graphing. If a system has no solution or infinitely many solutions, so state.
$$
\left\{\begin{array}{l}
{4 x+6 y=12} \\
{2 x+3 y=6}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 36

Solve each system by graphing. If a system has no solution or infinitely many solutions, so state.
$$
\left\{\begin{array}{l}
{2 x-y=0} \\
{2 y-4 x=0}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 37

Find the slope and the $y$ -intercept of the graph of each line in the system of equations. Then, use that information to determine the number of solutions of the system.
$$
\left\{\begin{array}{l}
{y=6 x-7} \\
{y=-2 x+1}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 38

Find the slope and the $y$ -intercept of the graph of each line in the system of equations. Then, use that information to determine the number of solutions of the system.
$$
\left\{\begin{array}{l}
{y=\frac{1}{2} x+8} \\
{y=4 x-10}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 39

Find the slope and the $y$ -intercept of the graph of each line in the system of equations. Then, use that information to determine the number of solutions of the system.
$$
\left\{\begin{array}{l}
{3 x-y=-3} \\
{y-3 x=3}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 40

Find the slope and the $y$ -intercept of the graph of each line in the system of equations. Then, use that information to determine the number of solutions of the system.
$$
\left\{\begin{array}{l}
{x+4 y=4} \\
{12 y=12-3 x}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 41

Find the slope and the $y$ -intercept of the graph of each line in the system of equations. Then, use that information to determine the number of solutions of the system.
$$
\left\{\begin{array}{l}
{x+y=6} \\
{x+y=8}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 42

Find the slope and the $y$ -intercept of the graph of each line in the system of equations. Then, use that information to determine the number of solutions of the system.
$$
\left\{\begin{array}{l}
{5 x+y=0} \\
{5 x+y=6}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 43

Find the slope and the $y$ -intercept of the graph of each line in the system of equations. Then, use that information to determine the number of solutions of the system.
$$
\left\{\begin{array}{l}
{6 x+y=0} \\
{2 x+2 y=0}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 44

Find the slope and the $y$ -intercept of the graph of each line in the system of equations. Then, use that information to determine the number of solutions of the system.
$$
\left\{\begin{array}{l}
{x+y=1} \\
{2 x-2 y=5}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 45

Use a graphing calculator to solve each system.
$$
\left\{\begin{array}{l}
{y=4-x} \\
{y=2+x}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 46

Use a graphing calculator to solve each system.
$$
\left\{\begin{array}{l}
{3 x-6 y=4} \\
{2 x+y=1}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 47

Use a graphing calculator to solve each system.
$$
\left\{\begin{array}{l}
{6 x-2 y=5} \\
{3 x=y+10}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 48

Use a graphing calculator to solve each system.
$$
\left\{\begin{array}{l}
{x-3 y=-2} \\
{5 x+y=10}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 49

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{y=3 x+6} \\
{y=-2 x-4}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 50

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{y=x+3} \\
{y=-2 x-3}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 51

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{2 y=3 x+2} \\
{3 x-2 y=6}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 52

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{3 x-6 y=18} \\
{x=2 y+3}
\end{array}\right.
$$

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Ian M.

Numerade Educator

Problem 53

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x+y=2} \\
{y=x-4}
\end{array}\right.
$$

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Ian M.

Numerade Educator

Problem 54

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x+y=1} \\
{y=x+5}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 55

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x=3} \\
{3 y=6-2 x}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 56

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x=4} \\
{2 y=12-4 x}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 57

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{y=\frac{3}{4} x+3} \\
{y=-\frac{x}{4}-1}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 58

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{y=\frac{2}{3} x+4} \\
{y=-\frac{x}{3}+7}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 59

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{y=-x-2} \\
{y=-3 x+6}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 60

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{y=2 x-4} \\
{y=-5 x+3}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 61

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{-x+3 y=-11} \\
{3 x-y=17}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 62

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{2 x-3 y=-18} \\
{3 x+2 y=-1}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 63

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x+y=2} \\
{y=x}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 64

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x+y=4} \\
{y=x}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 65

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{4 x-2 y=8} \\
{y=2 x-4}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 66

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{2 y=-6 x-12} \\
{3 x+y=-6}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 67

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x+4 y=-2} \\
{y=-x-5}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 68

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{3 x+2 y=-8} \\
{2 x-3 y=-1}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 69

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{y=-3} \\
{-x+2 y=-4}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 70

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{y=-4} \\
{-2 x-y=8}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 71

Solve each system of equations by graphing.
$$
\left\{\begin{array}{l}
{x+2 y=-4} \\
{x-\frac{1}{2} y=6}
\end{array}\right.
$$

IM

Ian M.

Numerade Educator

Problem 72

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

IM

Ian M.

Numerade Educator

Problem 73

Refer to the graph below. In what month did Facebook and MySpace have the same number of unique visitors? Estimate the number.
(GRAPH CANNOT COPY)

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Problem 74

Refer to the graph below. In what year was average number of gallons of milk and carbonated soft drinks consumed per person the same? Estimate the number of gallons.
(GRAPH CANNOT COPY)

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Numerade Educator

Problem 75

Latitude and Longitude. Refer to the following map.
a. Name three American cities that lie on a latitude line of $30^{\circ}$.
b. Name three American cities that lie on a longitude line of $-90^{\circ}$
c. What city lies on both lines?
(MAP CANNOT COPY)

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Problem 76

The following graph illustrates the law of supply and demand.
a. Complete each sentence with the word increases or decreases. As the price of an item increases, the supply of the item $\quad$ As the price of an item increases, the demand for the item
b. For what price will the supply equal the demand? How many items will be supplied for this price?
(GRAPH CANNOT COPY)

IM

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Problem 77

Refer to the graph below.
a. Which political candidate was ahead on October 28 and by how much?
b. On what day did the challenger pull even with the incumbent?
c. If the election was held November $4,$ who did the poll predict would win, and by how many percentage points?
(GRAPH CANNOT COPY)

IM

Ian M.

Numerade Educator

Problem 78

The equations describing the paths of two airplanes are $y=-\frac{1}{2} x+3$ and $3 y=2 x+2 .$ Graph each equation on the radar screen shown. Is there a possibility of a midair collision? If so, where?
(GRAPH CANNOT COPY)

IM

Ian M.

Numerade Educator

Problem 79

Photographers often use the rule of thirds to add more interest to a photo rather than simply centering the subject. They imagine two horizontal and two vertical lines dividing the image into a grid of 9 equal parts (like a tic-tac-toe game). The four points where the lines intersect are possible "sweet spots" for placing the subject of the picture. Give the coordinates of the four sweet spots for the image below. Is the sea gull properly placed?
(FIGURE CANNOT COPY)

IM

Ian M.

Numerade Educator

Problem 80

A television camera is located at $(-2,0)$ and will follow the launch of a space shuttle, as shown here. (Each unit in the illustration is 1 mile.) As the shuttle rises vertically on a path described by $x=2,$ the farthest the camera can tilt back is a line of sight given by $y=\frac{5}{2} x+5 .$ For how many miles of the shuttle's flight will it be in view of the camera?
(GRAPH CANNOT COPY)

IM

Ian M.

Numerade Educator

Problem 81

Explain why it is difficult to determine the solution of the system in the graph.
(GRAPH CANNOT COPY)

IM

Ian M.

Numerade Educator

Problem 82

Without graphing, how can you tell that the graphs of $y=2 x+1$ and $y=3 x+2$ intersect?

IM

Ian M.

Numerade Educator

Problem 83

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

IM

Ian M.

Numerade Educator

Problem 84

What is an inconsistent system? What are dependent equations?

IM

Ian M.

Numerade Educator

Problem 85

Suppose the graphs of the two linear equations of a system are the same line. What is wrong with the following statement? The system has infinitely many solutions. Any ordered pair is a
solution of the system.

IM

Ian M.

Numerade Educator

Problem 86

The graph below is for the month of January, $2010 .$ It shows the total number of doses of the HINI Swine Flu vaccine that had been produced to date. It also shows the total number of doses that U.S. health officials had ordered to date. Did the total number of doses ordered ever equal or surpass the total number of doses produced? How can you tell?
(GRAPH CANNOT COPY)

IM

Ian M.

Numerade Educator

Problem 87

Solve each inequality. Write the solution set in interval notation and graph it.
$$
-4(2 y+2) \leq 4 y+28
$$

IM

Ian M.

Numerade Educator

Problem 88

Solve each inequality. Write the solution set in interval notation and graph it.
$$
-5<3 t+4 \leq 13
$$

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Ian M.

Numerade Educator

Problem 89

Solve each inequality. Write the solution set in interval notation and graph it.
$$
-1 \leq-\frac{1}{2} n
$$

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Ian M.

Numerade Educator

Problem 90

Solve each inequality. Write the solution set in interval notation and graph it.
$$
\frac{1}{3}+\frac{c}{5}>-\frac{3}{2}
$$

IM

Ian M.

Numerade Educator

Problem 91

Can a system of two linear equations in two variables be inconsistent but have dependent equations? Explain.

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Problem 92

Construct a system of two linear equations that has a solution of $(-2,6)$

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Problem 93

Write a system of two linear equations such that $(2,3)$ is a solution of the first equation but is not a solution of the second equation.

IM

Ian M.

Numerade Educator

Problem 94

Write a system of two linear equations such that $(2,3)$ is a solution of the first equation but is not a solution of the second equation.

IM

Ian M.

Numerade Educator