Chapter 4 Review Exercises
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- Jan 6, 2020
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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 1
- (−1,−5)
- (−3,4)
- (2,−3)
- (1,52)
Exercise 2
- (4,3)
- (−4,3)
- (−4,−3)
- (4,−3)
- Answer
Exercise 3
- (−2,0)
- (0,−4)
- (0,5)
- (3,0)
Exercise 4
- (2,32)
- (3,43)
- (13,−4)
- (12,−5)
- Answer
Identify Points on a Graph
In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.
Exercise 5
Exercise 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 7
5x+y=10
- (5,1)
- (2,0)
- (4,−10)
Exercise 8
y=6x−2
- (1,4)
- (13,0)
- (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 9
y=4x−1
x | y | (x,y) |
0 | ||
1 | ||
-2 |
Exercise 10
y=−12x+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 thex- 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