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Graphing Linear Equations and Inequalities: Exercise Supplement

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

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Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information. The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. This module contains the exercise supplement for the chapter "Graphing Linear Equations and Inequalities in One and Two Variables".

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Exercise Supplement

Graphing Linear Equations and Inequalities in One Variable ((Reference))

For the following problems, graph the equations and inequalities.

Exercise 1

6x−18=6 6x−18=6
A horizontal line with arrows on both ends labeled as x.

Solution

x=4 x=4

A number line with arrows on each end, labeled from negative two to four in increments of one. There is a closed circle at four.

Exercise 2

4x−3=−7 4x−3=−7
A horizontal line with arrows on both ends labeled as x.

Exercise 3

5x−1=2 5x−1=2
A horizontal line with arrows on both ends labeled as x.

Solution

x= 3 5 x= 3 5

A number line with arrows on each end, labeled from negative one to ttwo in increments of one. There is a closed circle at three over five.

Exercise 4

10x−16<4 10x−16<4
A horizontal line with arrows on both ends labeled as x.

Exercise 5

−2y+1≤5 −2y+1≤5
A horizontal line with arrows on both ends labeled as y.

Solution

y≥−2 y≥−2

A number line with arrows on each end, labeled from negative three to three, in increments of one. There is a closed circle at negative two. A dark line is orginating from this circle and heading towards the right of negative two.

Exercise 6

−7a 12 ≥2 −7a 12 ≥2
A horizontal line with arrows on both ends labeled as a.

Exercise 7

3x+4≤12 3x+4≤12
A horizontal line with arrows on both ends labeled as x.

Solution

x≤ 8 3 x≤ 8 3

A number line with arrows on each end, labeled from negative two to three, in increments of one. There is a closed circle at a point between two and three. A dark line is orginating from this circle and heading towards the left of it.

Exercise 8

−16≤5x−1≤−11 −16≤5x−1≤−11
A horizontal line with arrows on both ends labeled as x.

Exercise 9

0<−3y+9≤9 0<−3y+9≤9
A horizontal line with arrows on both ends labeled as y.

Solution

0≤y<3 0≤y<3

A number line with arrows on each end, labeled from negative one to four, in increments of one. There is a closed circle at zero and an open circle at three. These circles are connected by a a black line.

Exercise 10

−5c 2 +1=7 −5c 2 +1=7
A horizontal line with arrows on both ends labeled as c.

Plotting Points in the Plane ((Reference))

Exercise 11

Draw a coordinate system and plot the following ordered pairs.

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

Solution



Total six points plotted in an xy-coordinate plane. The coordinates of these points are negative one, negative three; zero, three; three, one; three, zero; four, negative two; and five, negative two over three.

Exercise 12

As accurately as possible, state the coordinates of the points that have been plotted on the graph.
Total seven points plotted on an xy-plane. The coordinates of these points are one, three; two, one; three,zero; three, negative two; negative one, negative three; negative three, three.

Graphing Linear Equations in Two Variables ((Reference))

Exercise 13

What is the geometric structure of the graph of all the solutions to the linear equation y=4x−9 y=4x−9 ?

Solution

a straight line

Graphing Linear Equations in Two Variables ((Reference)) - Graphing Equations in Slope-Intercept Form ((Reference))

For the following problems, graph the equations.

Exercise 14

y−x=2 y−x=2

Exercise 15

y+x−3=0 y+x−3=0

Solution

A graph of a line passing through two points with coordinates zero, three and five, zero.

Exercise 16

−2x+3y=−6 −2x+3y=−6

Exercise 17

2y+x−8=0 2y+x−8=0

Solution

A graph of a line passing through two points with coordinates zero, four and eight, zero.

Exercise 18

4(x−y)=12 4(x−y)=12

Exercise 19

3y−4x+12=0 3y−4x+12=0

Solution

A graph of a line passing through two points with coordinates zero, three and negative four, zero.

Exercise 20

y=−3 y=−3

Exercise 21

y−2=0 y−2=0

Solution

A graph of a line parallel to x-axis in an xy plane. The line is labeled as ' y equals two'. The line crosses the y-axis at y equals two.

Exercise 22

x=4 x=4

Exercise 23

x+1=0 x+1=0

Solution

A graph of a line parallel to y-axis in an xy plane. The line is labeled as 'x equals negative one'. The line crosses the x-axis at x equals negative one.

Exercise 24

x=0 x=0

Exercise 25

y=0 y=0

Solution

A graph of a line in an xy plane coincident to x-axis labeled as 'y equals zero'.

The Slope-Intercept Form of a Line ((Reference))

Exercise 26

Write the slope-intercept form of a straight line.

Exercise 27

The slope of a straight line is a
of the steepness of the line.

Solution

measure

Exercise 28

Write the formula for the slope of a line that passes through the points ( x 1 ,y ) 1 ( x 1 ,y ) 1 and ( x 2 ,y ) 2 ( x 2 ,y ) 2 .

For the following problems, determine the slope and y-intercept y-intercept of the lines.

Exercise 29

y=4x+10 y=4x+10

Solution

slope:  4 y-intercept:  ( 0,10 ) slope:  4 y-intercept:  ( 0,10 )

Exercise 30

y=3x−11 y=3x−11

Exercise 31

y=9x−1 y=9x−1

Solution

slope:  9 y-intercept:  ( 0,−1 ) slope:  9 y-intercept:  ( 0,−1 )

Exercise 32

y=−x+2 y=−x+2

Exercise 33

y=−5x−4 y=−5x−4

Solution

slope:  −5 y-intercept:  ( 0,−4 ) slope:  −5 y-intercept:  ( 0,−4 )

Exercise 34

y=x y=x

Exercise 35

y=−6x y=−6x

Solution

slope:  −6 y-intercept:  ( 0,0 ) slope:  −6 y-intercept:  ( 0,0 )

Exercise 36

3y=4x+9 3y=4x+9

Exercise 37

4y=5x+1 4y=5x+1

Solution

slope:   5 4 y-intercept:  ( 0, 1 4 ) slope:   5 4 y-intercept:  ( 0, 1 4 )

Exercise 38

2y=9x 2y=9x

Exercise 39

5y+4x=6 5y+4x=6

Solution

slope: −  4 5 y-intercept:  ( 0, 6 5 ) slope: −  4 5 y-intercept:  ( 0, 6 5 )

Exercise 40

7y+3x=10 7y+3x=10

Exercise 41

6y−12x=24 6y−12x=24

Solution

slope:  2 y-intercept:  ( 0,4 ) slope:  2 y-intercept:  ( 0,4 )

Exercise 42

5y−10x−15=0 5y−10x−15=0

Exercise 43

3y+3x=1 3y+3x=1

Solution

slope:  −1 y-intercept:  ( 0, 1 3 ) slope:  −1 y-intercept:  ( 0, 1 3 )

Exercise 44

7y+2x=0 7y+2x=0

Exercise 45

y=4 y=4

Solution

slope:  0 y-intercept:  ( 0,4 ) slope:  0 y-intercept:  ( 0,4 )

For the following problems, find the slope, if it exists, of the line through the given pairs of points.

Exercise 46

(5,2), (6,3) (5,2), (6,3)

Exercise 47

(8,−2), (10,−6) (8,−2), (10,−6)

Solution

slope:  −2 slope:  −2

Exercise 48

(0,5), (3,4) (0,5), (3,4)

Exercise 49

(1,−4), (3,3) (1,−4), (3,3)

Solution

slope:   7 2 slope:   7 2

Exercise 50

(0,0), (−8,−5) (0,0), (−8,−5)

Exercise 51

(−6,1), (−2,7) (−6,1), (−2,7)

Solution

slope:   3 2 slope:   3 2

Exercise 52

(−3,−2), (−4,−5) (−3,−2), (−4,−5)

Exercise 53

(4,7), (4,−2) (4,7), (4,−2)

Solution

No Slope

Exercise 54

(−3,1), (4,1) (−3,1), (4,1)

Exercise 55

( 1 3 , 3 4 ),( 2 9 ,− 5 6 ) ( 1 3 , 3 4 ),( 2 9 ,− 5 6 )

Solution

slope:   57 4 slope:   57 4

Exercise 56

Moving left to right, lines with
slope rise while lines with
slope decline.

Exercise 57

Compare the slopes of parallel lines.

Solution

The slopes of parallel lines are equal.

Finding the Equation of a Line ((Reference))

For the following problems, write the equation of the line using the given information. Write the equation in slope-intercept form.

Exercise 58

Slope=4, y-intercept=5 Slope=4, y-intercept=5

Exercise 59

Slope=3, y-intercept=−6 Slope=3, y-intercept=−6

Solution

y=3x−6 y=3x−6

Exercise 60

Slope=1, y-intercept=8 Slope=1, y-intercept=8

Exercise 61

Slope=1, y-intercept=−2 Slope=1, y-intercept=−2

Solution

y=x−2 y=x−2

Exercise 62

Slope=−5, y-intercept=1 Slope=−5, y-intercept=1

Exercise 63

Slope=−11, y-intercept=−4 Slope=−11, y-intercept=−4

Solution

y=−11x−4 y=−11x−4

Exercise 64

Slope=2, y-intercept=0 Slope=2, y-intercept=0

Exercise 65

Slope=−1, y-intercept=0 Slope=−1, y-intercept=0

Solution

y=−x y=−x

Exercise 66

m=3, (4,1) m=3, (4,1)

Exercise 67

m=2, (1,5) m=2, (1,5)

Solution

y=2x+3 y=2x+3

Exercise 68

m=6, (5,−2) m=6, (5,−2)

Exercise 69

m=−5, (2,−3) m=−5, (2,−3)

Solution

y=−5x+7 y=−5x+7

Exercise 70

m=−9, (−4,−7) m=−9, (−4,−7)

Exercise 71

m=−2, (0,2) m=−2, (0,2)

Solution

y=−2x+2 y=−2x+2

Exercise 72

m=−1, (2,0) m=−1, (2,0)

Exercise 73

(2,3), (3,5) (2,3), (3,5)

Solution

y=2x−1 y=2x−1

Exercise 74

(4,4), (5,1) (4,4), (5,1)

Exercise 75

(6,1), (5,3) (6,1), (5,3)

Solution

y=−2x+13 y=−2x+13

Exercise 76

(8,6), (7,2) (8,6), (7,2)

Exercise 77

(−3,1), (2,3) (−3,1), (2,3)

Solution

y= 2 5 x+ 11 5 y= 2 5 x+ 11 5

Exercise 78

(−1,4), (−2,−4) (−1,4), (−2,−4)

Exercise 79

(0,−5), (6,−1) (0,−5), (6,−1)

Solution

y= 2 3 x−5 y= 2 3 x−5

Exercise 80

(2,1), (6,1) (2,1), (6,1)

Exercise 81

(−5,7), (−2,7) (−5,7), (−2,7)

Solution

y=7 ( zero slope ) y=7 ( zero slope )

Exercise 82

(4,1), (4,3) (4,1), (4,3)

Exercise 83

(−1,−1), (−1,5) (−1,−1), (−1,5)

Solution

x=−1 ( no slope ) x=−1 ( no slope )

Exercise 84

(0,4), (0,−3) (0,4), (0,−3)

Exercise 85

(0,2), (1,0) (0,2), (1,0)

Solution

y=−2x+2 y=−2x+2

For the following problems, reading only from the graph, determine the equation of the line.

Exercise 86

A graph of a line sloped up and to the right. The line crosses the y-axis at y equals one, and crosses the x-axis at x equals negative two.

Exercise 87

A graph of a line sloped up and to the right. The line crosses the x-axis at x equals three, and crosses the y-axis at y equals negative two.

Solution

y= 2 3 x−2 y= 2 3 x−2

Exercise 88

A graph of a line sloped down and to the right. The line crosses the y-axis at y equals one, and crosses the x-axis at x equals four.

Exercise 89

A graph of a parallel to x-axis. The line crosses the y-axis at y equals negative two.

Solution

y=−2 y=−2

Exercise 90

A graph of a parallel to y-axis. The line crosses the x-axis at x equals three.

Exercise 91

A graph of a parallel to x-axis. The line crosses the y-axis at y equals one.

Solution

y=1 y=1

Graphing Linear Inequalities in Two Variables ((Reference))

For the following problems, graph the inequalities.

Exercise 92

y≤x+2 y≤x+2
An xy-plane with gridlines, labeled negative five and five on the both axes.

Exercise 93

y<− 1 2 x+3 y<− 1 2 x+3
An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

A line in an xy plane passing through two points with coordinates zero, three and four, one. The region below the line is shaded.

Exercise 94

y> 1 3 x−3 y> 1 3 x−3
An xy-plane with gridlines, labeled negative five and five on the both axes.

Exercise 95

−2x+3y≤−6 −2x+3y≤−6
An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

A line in an xy plane passing through two points with coordinates zero, negative two and three, zero. The region below the line is shaded.

Exercise 96

2x+5y≥20 2x+5y≥20
An xy-plane with gridlines, labeled negative five and five on the both axes.

Exercise 97

4x−y+12>0 4x−y+12>0
An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

A line in an xy plane passing through two points with coordinates zero, twelve and three, zero. The region to the right of the line is shaded.

Exercise 98

y≥−2 y≥−2
An xy-plane with gridlines, labeled negative five and five on the both axes.

Exercise 99

x<3 x<3
An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

A dashed line parallel to y-axis in an xy plane. The line crosses the x-axis at x equals three. The region to the left of the line is shaded.

Exercise 100

y≤0 y≤0
An xy-plane with gridlines, labeled negative five and five on the both axes.

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