# Chapter 4 Review Exercises

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- Lynn Marecek
- Professor (Mathematics) at Santa Ana College
- Publisher: OpenStax CNX

## Chapter 4 Review Exercises

__Rectangular Coordinate System__

**Plot Points in a Rectangular Coordinate System**

In the following exercises, plot each point in a rectangular coordinate system.

##### Exercise \(\PageIndex{1}\)

- (−1,−5)
- (−3,4)
- (2,−3)
- \(\left(1, \frac{5}{2}\right)\)

##### Exercise \(\PageIndex{2}\)

- (4,3)
- (−4,3)
- (−4,−3)
- (4,−3)

**Answer**

##### Exercise \(\PageIndex{3}\)

- (−2,0)
- (0,−4)
- (0,5)
- (3,0)

##### Exercise \(\PageIndex{4}\)

- \(\left(2, \frac{3}{2}\right)\)
- \(\left(3, \frac{4}{3}\right)\)
- \(\left(\frac{1}{3},-4\right)\)
- \(\left(\frac{1}{2},-5\right)\)

**Answer**

**Identify Points on a Graph**

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

##### Exercise \(\PageIndex{5}\)

##### Exercise \(\PageIndex{6}\)

**Answer**-
a. (2,0)

b (0,−5)

c (−4.0)

d (0,3)

**Verify Solutions to an Equation in Two Variables**

In the following exercises, which ordered pairs are solutions to the given equations?

##### Exercise \(\PageIndex{7}\)

\(5x+y=10\)

- (5,1)
- (2,0)
- (4,−10)

##### Exercise \(\PageIndex{8}\)

\(y=6x−2\)

- (1,4)
- \(\left(\frac{1}{3}, 0\right)\)
- (6,−2)

**Answer**-
1, 2

**Complete a Table of Solutions to a Linear Equation in Two Variables**

In the following exercises, complete the table to find solutions to each linear equation.

##### Exercise \(\PageIndex{9}\)

\(y=4 x-1\)

x |
y |
(x,y) |

0 | ||

1 | ||

-2 |

##### Exercise \(\PageIndex{10}\)

\(y=-\frac{1}{2} x+3\)

x |
y |
(x,y) |

0 | ||

4 | ||

-2 |

**Answer**-
**x****y****(x,y)**0 3 (0,3) 4 1 (4, 1) −2 4 (−2,4)

##### Exercise \(\PageIndex{11}\)

\(x+2 y=5\)

x |
y |
(x,y) |

0 | ||

1 | ||

-1 |

##### Exercise \(\PageIndex{12}\)

\(3x+2y=6\)

x | y | (x,y) |

0 | ||

0 | ||

-2 |

**Answer**-
x y (x,y) 0 −3 (0,−3) 2 0 (2,0) −2 −6 (−2,−6)

**Find Solutions to a Linear Equation in Two Variables**

In the following exercises, find three solutions to each linear equation.

##### Exercise \(\PageIndex{13}\)

\(x+y=3\)

##### Exercise \(\PageIndex{14}\)

\(x+y=-4\)

**Answer**-
Answers will vary.

##### Exercise \(\PageIndex{15}\)

\(y=3 x+1\)

##### Exercise \(\PageIndex{16}\)

\(y=-x-1\)

**Answer**-
Answers will vary.

__Graphing Linear Equations__

**Recognize the Relation Between the Solutions of an Equation and its Graph**

In the following exercises, for each ordered pair, decide:

- Is the ordered pair a solution to the equation?
- Is the point on the line?

##### Exercise \(\PageIndex{17}\)

\(y=−x+4\)

(0,4) (−1,3)

(2,2) (−2,6)

##### Exercise \(\PageIndex{18}\)

\(y=\frac{2}{3} x-1\)

\((0,-1) (3,1)\)

\((-3,-3) (6,4)\)

**Answer**-
- yes; yes
- yes; no

**Graph a Linear Equation by Plotting Points**

In the following exercises, graph by plotting points.

##### Exercise \(\PageIndex{19}\)

\(y=4x-3\)

##### Exercise \(\PageIndex{20}\)

\(y=-3x\)

**Answer**

##### Exercise \(\PageIndex{21}\)

\(y=\frac{1}{2} x+3\)

##### Exercise \(\PageIndex{22}\)

\(x-y=6\)

**Answer**

##### Exercise \(\PageIndex{23}\)

\(2x+y=7\)

##### Exercise \(\PageIndex{24}\)

\(3x-2y=6\)

**Answer**

**Graph Vertical and Horizontal lines**

In the following exercises, graph each equation.

##### Exercise \(\PageIndex{25}\)

\(y=-2\)

##### Exercise \(\PageIndex{26}\)

\(x=3\)

**Answer**

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

##### Exercise \(\PageIndex{27}\)

\(y=-2 x\) and \(y=-2\)

##### Exercise \(\PageIndex{28}\)

\(y=\frac{4}{3} x\) and \(y=\frac{4}{3}\)

**Answer**

__Graphing with Intercepts__

**Identify the \(x\)- and \(y\)-Intercepts on a Graph**

In the following exercises, find the \(x\)- and \(y\)-intercepts.

##### Exercise \(\PageIndex{29}\)

##### Exercise \(\PageIndex{30}\)

**Answer**-
\((3,0)\) and \((0,3)\)

**Find the\(x\)- and \(y\)-Intercepts from an Equation of a Line**

In the following exercises, find the intercepts of each equation.

##### Exercise \(\PageIndex{31}\)

\(x+y=5\)

##### Exercise \(\PageIndex{32}\)

\(x-y=-1\)

**Answer**-
\((-1,0),(0,1)\)

##### Exercise \(\PageIndex{33}\)

\(x+2y=6\)

##### Exercise \(\PageIndex{34}\)

\(2x+3y=12\)

**Answer**-
\((6,0),(0,4)\)

##### Exercise \(\PageIndex{35}\)

\(y=\frac{3}{4} x-12\)

##### Exercise \(\PageIndex{36}\)

\(y=3x\)

**Answer**-
\((0,0)\)

**Graph a Line Using the Intercepts**

In the following exercises, graph using the intercepts.

##### Exercise \(\PageIndex{37}\)

\(-x+3y=3\)

##### Exercise \(\PageIndex{38}\)

\(x+y=-2\)

**Answer**

##### Exercise \(\PageIndex{39}\)

\(x-y=4\)

##### Exercise \(\PageIndex{40}\)

\(2x-y=5\)

**Answer**

##### Exercise \(\PageIndex{41}\)

\(2x-4y=8\)

##### Exercise \(\PageIndex{42}\)

\(y=2x\)

**Answer**

__Slope of a Line__

**Use Geoboards to Model Slope**

In the following exercises, find the slope modeled on each geoboard.

##### Exercise \(\PageIndex{43}\)

##### Exercise \(\PageIndex{44}\)

**Answer**-
\(\frac{4}{3}\)

##### Exercise \(\PageIndex{45}\)

##### Exercise \(\PageIndex{46}\)

**Answer**-
\(-\frac{2}{3}\)

##### Exercise \(\PageIndex{47}\)

\(\frac{1}{3}\)

##### Exercise \(\PageIndex{48}\)

\(\frac{3}{2}\)

**Answer**

##### Exercise \(\PageIndex{49}\)

\(-\frac{2}{3}\)

##### Exercise \(\PageIndex{50}\)

\(-\frac{1}{2}\)

**Answer**

**Use **\(m=\frac{\text { rise }}{\text { run }}\) **to find the Slope of a Line from its Graph**

In the following exercises, find the slope of each line shown.

##### Exercise \(\PageIndex{51}\)

##### Exercise \(\PageIndex{52}\)

**Answer**-
1

##### Exercise \(\PageIndex{53}\)

##### Exercise \(\PageIndex{54}\)

**Answer**-
\(-\frac{1}{2}\)

**Find the Slope of Horizontal and Vertical Lines**

In the following exercises, find the slope of each line.

##### Exercise \(\PageIndex{55}\)

\(y=2\)

##### Exercise \(\PageIndex{56}\)

\(x=5\)

**Answer**-
undefined

##### Exercise \(\PageIndex{57}\)

\(x=-3\)

##### Exercise \(\PageIndex{58}\)

\(y=-1\)

**Answer**-
0

**Use the Slope Formula to find the Slope of a Line between Two Points**

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

##### Exercise \(\PageIndex{59}\)

\((-1,-1),(0,5)\)

##### Exercise \(\PageIndex{60}\)

\((3,5),(4,-1)\)

**Answer**-
−6

##### Exercise \(\PageIndex{61}\)

\((-5,-2),(3,2)\)

##### Exercise \(\PageIndex{62}\)

\((2,1),(4,6)\)

**Answer**-
\(\frac{5}{2}\)

**Graph a Line Given a Point and the Slope**

*In the following exercises, graph each line with the given point and slope.*

##### Exercise \(\PageIndex{63}\)

\((2,-2) ; \quad m=\frac{5}{2}\)

##### Exercise \(\PageIndex{64}\)

\((-3,4) ; \quad m=-\frac{1}{3}\)

**Answer**

##### Exercise \(\PageIndex{65}\)

\(x\) -intercept \(-4 ; \quad m=3\)

##### Exercise \(\PageIndex{66}\)

\(y\) -intercept \(1 ; \quad m=-\frac{3}{4}\)

**Answer**

**Solve Slope Applications**

In the following exercises, solve these slope applications.

##### Exercise \(\PageIndex{67}\)

The roof pictured below has a rise of \(10\) feet and a run of \(15\) feet. What is its slope?

##### Exercise \(\PageIndex{68}\)

A mountain road rises \(50\) feet for a \(500\)-foot run. What is its slope?

**Answer**-
\(\frac{1}{10}\)

__Intercept Form of an Equation of a Line__

**Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line**

In the following exercises, use the graph to find the slope and *y*-intercept of each line. Compare the values to the equation \(y=mx+b\).

##### Exercise \(\PageIndex{69}\)

\(y=4x−1\)

##### Exercise \(\PageIndex{70}\)

\(y=-\frac{2}{3} x+4\)

**Answer**-
slope \(m=-\frac{2}{3}\) and \(y\)-intercept \((0,4)\)

**Identify the Slope and y-Intercept from an Equation of a Line**

In the following exercises, identify the slope and \(y\)-intercept of each line.

##### Exercise \(\PageIndex{71}\)

\(y=-4 x+9\)

##### Exercise \(\PageIndex{72}\)

\(y=\frac{5}{3} x-6\)

**Answer**-
\(\frac{5}{3} ;(0,-6)\)

##### Exercise \(\PageIndex{73}\)

\(5x+y=10\)

##### Exercise \(\PageIndex{74}\)

\(4x-5y=8\)

**Answer**-
\(\frac{4}{5} ;\quad \left(0,-\frac{8}{5}\right)\)

**Graph a Line Using Its Slope and Intercept**

In the following exercises, graph the line of each equation using its slope and \(y\)-intercept.

##### Exercise \(\PageIndex{75}\)

\(y=2x+3\)

##### Exercise \(\PageIndex{76}\)

\(y=-x-1\)

**Answer**

##### Exercise \(\PageIndex{77}\)

\(y=-\frac{2}{5} x+3\)

##### Exercise \(\PageIndex{78}\)

\(4x-3y=12\)

**Answer**

In the following exercises, determine the most convenient method to graph each line.

##### Exercise \(\PageIndex{79}\)

\(x=5\)

##### Exercise \(\PageIndex{80}\)

\(y=-3\)

**Answer**-
horizontal line

##### Exercise \(\PageIndex{81}\)

\(2x+y=5\)

##### Exercise \(\PageIndex{82}\)

\(x-y=2\)

**Answer**-
intercepts

##### Exercise \(\PageIndex{83}\)

\(y=x+2\)

##### Exercise \(\PageIndex{84}\)

\(y=\frac{3}{4} x-1\)

**Answer**-
plotting points

**Graph and Interpret Applications of Slope–Intercept**

##### Exercise \(\PageIndex{85}\)

Katherine is a private chef. The equation \(C=6.5m+42\) models the relation between her weekly cost, \(C\), in dollars and the number of meals, \(m\), that she serves.

- Find Katherine’s cost for a week when she serves no meals.
- Find the cost for a week when she serves \(14\) meals.
- Interpret the slope and \(C\)-intercept of the equation.
- Graph the equation.

##### Exercise \(\PageIndex{86}\)

Marjorie teaches piano. The equation \(P=35h−250\) models the relation between her weekly profit, \(P\), in dollars and the number of student lessons, \(s\), that she teaches.

- Find Marjorie’s profit for a week when she teaches no student lessons.
- Find the profit for a week when she teaches \(20\) student lessons.
- Interpret the slope and \(P\)-intercept of the equation.
- Graph the equation.

**Answer**-
- \(−$250\)
- \($450\)
- The slope, \(35\), means that Marjorie’s weekly profit, \(P\), increases by \($35\) for each additional student lesson she teaches. The \(P\)-intercept means that when the number of lessons is \(0\), Marjorie loses \($250\).

**Use Slopes to Identify Parallel Lines**

In the following exercises, use slopes and \(y\)-intercepts to determine if the lines are parallel.

##### Exercise \(\PageIndex{87}\)

\(4x-3y=-1 ; \quad y=\frac{4}{3} x-3\)

##### Exercise \(\PageIndex{88}\)

\(2 x-y=8 ; \quad x-2 y=4\)

**Answer**-
not parallel

**Use Slopes to Identify Perpendicular Lines**

In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

##### Exercise \(\PageIndex{89}\)

\(y=5x-1 ; \quad 10x+2y=0\)

##### Exercise \(\PageIndex{90}\)

\(3x-2y=5 ; \quad 2x+3y=6\)

**Answer**-
perpendicular

__Find the Equation of a Line__

**Find an Equation of the Line Given the Slope and y-Intercept**

In the following exercises, find the equation of a line with given slope and \(y\)-intercept. Write the equation in slope–intercept form.

##### Exercise \(\PageIndex{91}\)

slope \(\frac{1}{3}\) and \(y\)-intercept \((0,-6)\)

##### Exercise \(\PageIndex{92}\)

slope \(-5\) and \(y\)-intercept \((0,-3)\)

**Answer**-
\(y=-5x-3\)

##### Exercise \(\PageIndex{93}\)

slope \(0\) and \(y\)-intercept \((0,4)\)

##### Exercise \(\PageIndex{94}\)

slope \(-2\) and \(y\)-intercept \((0,0)\)

**Answer**-
\(y=-2x\)

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

##### Exercise \(\PageIndex{95}\)

##### Exercise \(\PageIndex{96}\)

**Answer**-
\(y=-3x+5\)

##### Exercise \(\PageIndex{97}\)

##### Exercise \(\PageIndex{98}\)

**Answer**-
\(y=-4\)

**Find an Equation of the Line Given the Slope and a Point**

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

##### Exercise \(\PageIndex{99}\)

\(m=-\frac{1}{4},\) point \((-8,3)\)

##### Exercise \(\PageIndex{100}\)

\(m=\frac{3}{5},\) point \((10,6)\)

**Answer**-
\(y=\frac{3}{5} x\)

##### Exercise \(\PageIndex{101}\)

Horizontal line containing \((-2,7)\)

##### Exercise \(\PageIndex{102}\)

\(m=-2,\) point \((-1,-3)\)

**Answer**-
\(y=-2x-5\)

**Find an Equation of the Line Given Two Points**

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.

##### Exercise \(\PageIndex{103}\)

\((2,10)\) and \((-2,-2)\)

##### Exercise \(\PageIndex{104}\)

\((7,1)\) and \((5,0)\)

**Answer**-
\(y=\frac{1}{2} x-\frac{5}{2}\)

##### Exercise \(\PageIndex{105}\)

\((3,8)\) and \((3,-4)\)

##### Exercise \(\PageIndex{106}\)

\((5,2)\) and \((-1,2)\)

**Answer**-
\(y=2\)

**Find an Equation of a Line Parallel to a Given Line**

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

##### Exercise \(\PageIndex{107}\)

line \(y=-3x+6,\) point \((1,-5)\)

##### Exercise \(\PageIndex{108}\)

line \(2x+5y=-10,\) point \((10,4)\)

**Answer**-
\(y=-\frac{2}{5} x+8\)

##### Exercise \(\PageIndex{109}\)

line \(x=4,\) point \((-2,-1)\)

##### Exercise \(\PageIndex{110}\)

line \(y=-5,\) point \((-4,3)\)

**Answer**-
\(y=3\)

**Find an Equation of a Line Perpendicular to a Given Line**

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

##### Exercise \(\PageIndex{111}\)

line \(y=-\frac{4}{5} x+2,\) point \((8,9)\)

##### Exercise \(\PageIndex{112}\)

line \(2x-3y=9,\) point \((-4,0)\)

**Answer**-
\(y=-\frac{3}{2} x-6\)

##### Exercise \(\PageIndex{113}\)

line \(y=3,\) point \((-1,-3)\)

##### Exercise \(\PageIndex{114}\)

line \(x=-5\) point \((2,1)\)

**Answer**-
\(y=1\)

__Graph Linear Inequalities__

**Verify Solutions to an Inequality in Two Variables**

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

##### Exercise \(\PageIndex{115}\)

Determine whether each ordered pair is a solution to the inequality \(y<x−3\):

- \((0,1)\)
- \((−2,−4)\)
- \((5,2)\)
- \((3,−1)\)
- \((−1,−5)\)

##### Exercise \(\PageIndex{116}\)

Determine whether each ordered pair is a solution to the inequality \(x+y>4\):

- \((6,1)\)
- \((−3,6)\)
- \((3,2)\)
- \((−5,10)\)
- \((0,0)\)

**Answer**-
- yes
- no
- yes
- yes
- no

**Recognize the Relation Between the Solutions of an Inequality and its Graph**

In the following exercises, write the inequality shown by the shaded region.

##### Exercise \(\PageIndex{117}\)

Write the inequality shown by the graph with the boundary line \(y=−x+2\).

##### Exercise \(\PageIndex{118}\)

Write the inequality shown by the graph with the boundary line \(y=\frac{2}{3} x-3\)

**Answer**-
\(y>\frac{2}{3} x-3\)

##### Exercise \(\PageIndex{119}\)

Write the inequality shown by the shaded region in the graph with the boundary line \(x+y=−4\).

##### Exercise \(\PageIndex{120}\)

Write the inequality shown by the shaded region in the graph with the boundary line \(x−2y=6\).

**Answer**-
\(x-2 y \geq 6\)

**Graph Linear Inequalities**

In the following exercises, graph each linear inequality.

##### Exercise \(\PageIndex{121}\)

Graph the linear inequality \(y>\frac{2}{5} x-4\)

##### Exercise \(\PageIndex{122}\)

Graph the linear inequality \(y \leq-\frac{1}{4} x+3\)

**Answer**

##### Exercise \(\PageIndex{123}\)

Graph the linear inequality \(x-y \leq 5\)

##### Exercise \(\PageIndex{124}\)

Graph the linear inequality \(3 x+2 y>10\)

**Answer**

##### Exercise \(\PageIndex{125}\)

Graph the linear inequality \(y \leq-3 x\)

##### Exercise \(\PageIndex{126}\)

Graph the linear inequality \(y<6\)

**Answer**

## Practice Test

##### Exercise \(\PageIndex{1}\)

Plot each point in a rectangular coordinate system.

- \((2,5)\)
- \((−1,−3)\)
- \((0,2)\)
- \(\left(-4, \frac{3}{2}\right)\)
- \((5,0)\)

##### Exercise \(\PageIndex{2}\)

Which of the given ordered pairs are solutions to the equation \(3x−y=6\)?

- \((3,3)\)
- \((2,0)\)
- \((4,−6)\)

**Answer**-
- yes
- yes
- no

##### Exercise \(\PageIndex{3}\)

Find three solutions to the linear equation \(y=-2x-4\)

##### Exercise \(\PageIndex{4}\)

Find the \(x\)- and \(y\)-intercepts of the equation \(4x-3y=12\)

**Answer**-
\((3,0),(0,-4)\)

Find the slope of each line shown.

##### Exercise \(\PageIndex{5}\)

##### Exercise \(\PageIndex{6}\)

**Answer**-
undefined

##### Exercise \(\PageIndex{7}\)

##### Exercise \(\PageIndex{8}\)

Find the slope of the line between the points \((5,2)\) and \((-1,-4)\)

**Answer**-
1

##### Exercise \(\PageIndex{9}\)

Graph the line with slope \(\frac{1}{2}\) containing the point \((-3,-4)\)

Graph the line for each of the following equations.

##### Exercise \(\PageIndex{10}\)

\(y=\frac{5}{3} x-1\)

**Answer**

##### Exercise \(\PageIndex{11}\)

\(y=-x\)

##### Exercise \(\PageIndex{12}\)

\(x-y=2\)

**Answer**

##### Exercise \(\PageIndex{13}\)

\(4x+2y=-8\)

##### Exercise \(\PageIndex{14}\)

\(y=2\)

**Answer**

##### Exercise \(\PageIndex{15}\)

\(x=-3\)

Find the equation of each line. Write the equation in slope–intercept form.

##### Exercise \(\PageIndex{16}\)

slope \(-\frac{3}{4}\) and \(y\)-intercept \((0,-2)\)

**Answer**-
\(y=-\frac{3}{4} x-2\)

##### Exercise \(\PageIndex{17}\)

\(m=2,\) point \((-3,-1)\)

##### Exercise \(\PageIndex{18}\)

containing \((10,1)\) and \((6,-1)\)

**Answer**-
\(y=\frac{1}{2} x-4\)

##### Exercise \(\PageIndex{19}\)

parallel to the line \(y=-\frac{2}{3} x-1,\) containing the point \((-3,8)\)

##### Exercise \(\PageIndex{20}\)

perpendicular to the line \(y=\frac{5}{4} x+2,\) containing the point \((-10,3)\)

**Answer**-
\(y=-\frac{4}{5} x-5\)

##### Exercise \(\PageIndex{21}\)

Write the inequality shown by the graph with the boundary line \(y=−x−3\).

Graph each linear inequality.

##### Exercise \(\PageIndex{22}\)

\(y>\frac{3}{2} x+5\)

**Answer**

##### Exercise \(\PageIndex{23}\)

\(x-y \geq-4\)

##### Exercise \(\PageIndex{24}\)

\(y \leq-5 x\)

**Answer**

##### Exercise \(\PageIndex{1}\)

\(y<3\)