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Elementary Algebra 2e

Review Exercises

Elementary Algebra 2eReview Exercises
Table of contents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Review Exercises

Rectangular Coordinate System

Plot Points in a Rectangular Coordinate System

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

556.
  1. ⓐ (−1,−5)(−1,−5)
  2. ⓑ (−3,4)(−3,4)
  3. ⓒ (2,−3)(2,−3)
  4. ⓓ (1,52)(1,52)
557.
  1. ⓐ (4,3)(4,3)
  2. ⓑ (−4,3)(−4,3)
  3. ⓒ (−4,−3)(−4,−3)
  4. ⓓ (4,−3)(4,−3)
558.
  1. ⓐ (−2,0)(−2,0)
  2. ⓑ (0,−4)(0,−4)
  3. ⓒ (0,5)(0,5)
  4. ⓓ (3,0)(3,0)
559.
  1. ⓐ (2,32)(2,32)
  2. ⓑ (3,43)(3,43)
  3. ⓒ (13,−4)(13,−4)
  4. ⓓ (12,−5)(12,−5)

Identify Points on a Graph

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

560.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (5, 3) is plotted and labeled "a". The point (2, negative 1) is plotted and labeled "b". The point (negative 3, negative 2) is plotted and labeled "c". The point (negative 1, 4) is plotted and labeled “d”.
561.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (2, 0) is plotted and labeled "a". The point (0, negative 5) is plotted and labeled "b". The point (negative 4, 0) is plotted and labeled "c". The point (0, 3) is plotted and labeled “d”.

Verify Solutions to an Equation in Two Variables

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

562.

5 x + y = 10 5 x + y = 10

  1. ⓐ (5,1)(5,1)
  2. ⓑ (2,0)(2,0)
  3. ⓒ (4,−10)(4,−10)
563.

y = 6 x − 2 y = 6 x − 2

  1. ⓐ (1,4)(1,4)
  2. ⓑ (13,0)(13,0)
  3. ⓒ (6,−2)(6,−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.

564.

y = 4 x − 1 y = 4 x − 1

xx yy (x,y)(x,y)
0
1
−2−2
565.

y = − 1 2 x + 3 y = − 1 2 x + 3

xx yy (x,y)(x,y)
0
4
−2−2
566.

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

xx yy (x,y)(x,y)
0
1
−1−1
567.

3 x + 2 y = 6 3 x + 2 y = 6

xx yy (x,y)(x,y)
0
0
−2−2

Find Solutions to a Linear Equation in Two Variables

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

568.

x + y = 3 x + y = 3

569.

x + y = −4 x + y = −4

570.

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

571.

y = − x − 1 y = − x − 1

Graphing Linear Equations

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

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

  1. ⓐ Is the ordered pair a solution to the equation?
  2. ⓑ Is the point on the line?
572.

y = − x + 4 y = − x + 4

ⓐ (0,4)(0,4)

ⓑ (−1,3)(−1,3)

ⓒ (2,2)(2,2)

ⓓ (−2,6)(−2,6)

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 4 is plotted as an arrow extending from the top left toward the bottom right.
573.

y = 2 3 x − 1 y = 2 3 x − 1

ⓐ (0,−1)(0,−1)

ⓑ (3,1)(3,1)

ⓒ (−3,−3)(−3,−3)

ⓓ (6,4)(6,4)

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 1 is plotted as an arrow extending from the bottom left toward the top right.

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

574.

y = 4 x − 3 y = 4 x − 3

575.

y = −3 x y = −3 x

576.

y = 1 2 x + 3 y = 1 2 x + 3

577.

x − y = 6 x − y = 6

578.

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

579.

3 x − 2 y = 6 3 x − 2 y = 6

Graph Vertical and Horizontal lines

In the following exercises, graph each equation.

580.

y = −2 y = −2

581.

x = 3 x = 3

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

582.

y=−2xy=−2x and y=−2y=−2

583.

y=43xy=43x and y=43y=43

Graphing with Intercepts

Identify the x- and y-Intercepts on a Graph

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

584.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.
585.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (3, 0) and (0, 3) is plotted.

Find the x- and y-Intercepts from an Equation of a Line

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

586.

x + y = 5 x + y = 5

587.

x − y = −1 x − y = −1

588.

x + 2 y = 6 x + 2 y = 6

589.

2 x + 3 y = 12 2 x + 3 y = 12

590.

y = 3 4 x − 12 y = 3 4 x − 12

591.

y = 3 x y = 3 x

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

592.

− x + 3 y = 3 − x + 3 y = 3

593.

x + y = −2 x + y = −2

594.

x − y = 4 x − y = 4

595.

2 x − y = 5 2 x − y = 5

596.

2 x − 4 y = 8 2 x − 4 y = 8

597.

y = 2 x y = 2 x

Slope of a Line

Use Geoboards to Model Slope

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

598.
The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.
599.
The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.
600.
The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.
601.
The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.

In the following exercises, model each slope. Draw a picture to show your results.

602.

1 3 1 3

603.

3 2 3 2

604.

− 2 3 − 2 3

605.

− 1 2 − 1 2

Use m=riserunm=riserun to find the Slope of a Line from its Graph

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

606.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 1, 3), (0, 0), and (1, negative 3) is plotted.
607.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.
608.
609.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 6) and (5, 2) is plotted.

Find the Slope of Horizontal and Vertical Lines

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

610.

y = 2 y = 2

611.

x = 5 x = 5

612.

x = −3 x = −3

613.

y = −1 y = −1

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.

614.

( −1 , −1 ) , ( 0 , 5 ) ( −1 , −1 ) , ( 0 , 5 )

615.

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

616.

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

617.

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

Graph a Line Given a Point and the Slope

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

618.

(2,−2)(2,−2); m=52m=52

619.

(−3,4)(−3,4); m=−13m=−13

620.

x-intercept −4−4; m=3m=3

621.

y-intercept 1; m=−34m=−34

Solve Slope Applications

In the following exercises, solve these slope applications.

622.

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

The figure shows a person on a ladder using a hammer on the roof of a building.
623.

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

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+by=mx+b.

624.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 4 x minus 1 is plotted from the lower left to the top right.

y = 4 x − 1 y = 4 x − 1

625.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x plus 4 is plotted from the top left to the bottom right.

y = − 2 3 x + 4 y = − 2 3 x + 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.

626.

y = −4 x + 9 y = −4 x + 9

627.

y = 5 3 x − 6 y = 5 3 x − 6

628.

5 x + y = 10 5 x + y = 10

629.

4 x − 5 y = 8 4 x − 5 y = 8

Graph a Line Using Its Slope and Intercept

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

630.

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

631.

y = − x − 1 y = − x − 1

632.

y = − 2 5 x + 3 y = − 2 5 x + 3

633.

4 x − 3 y = 12 4 x − 3 y = 12

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

634.

x = 5 x = 5

635.

y = −3 y = −3

636.

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

637.

x − y = 2 x − y = 2

638.

y = x + 2 y = x + 2

639.

y = 3 4 x − 1 y = 3 4 x − 1

Graph and Interpret Applications of Slope–Intercept

640.

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

  1. ⓐ Find Katherine’s cost for a week when she serves no meals.
  2. ⓑ Find the cost for a week when she serves 14 meals.
  3. ⓒ Interpret the slope and C-intercept of the equation.
  4. ⓓ Graph the equation.
641.

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

  1. ⓐ Find Marjorie’s profit for a week when she teaches no student lessons.
  2. ⓑ Find the profit for a week when she teaches 20 student lessons.
  3. ⓒ Interpret the slope and P–intercept of the equation.
  4. ⓓ Graph the equation.

Use Slopes to Identify Parallel Lines

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

642.

4 x − 3 y = −1 ; y = 4 3 x − 3 4 x − 3 y = −1 ; y = 4 3 x − 3

643.

2 x − y = 8 ; x − 2 y = 4 2 x − y = 8 ; x − 2 y = 4

Use Slopes to Identify Perpendicular Lines

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

644.

y = 5 x − 1 ; 10 x + 2 y = 0 y = 5 x − 1 ; 10 x + 2 y = 0

645.

3 x − 2 y = 5 ; 2 x + 3 y = 6 3 x − 2 y = 5 ; 2 x + 3 y = 6

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.

646.

slope 1313 and y-intercepty-intercept (0,−6)(0,−6)

647.

slope −5−5 and y-intercepty-intercept (0,−3)(0,−3)

648.

slope 00 and y-intercepty-intercept (0,4)(0,4)

649.

slope −2−2 and y-intercepty-intercept (0,0)(0,0)

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

650.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 x plus 1 is plotted from the bottom left to the top right.
651.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 3 x plus 5 is plotted from the top left to the bottom right.
652.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals three-fourths x minus 2 is plotted from the bottom left to the top right.
653.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 4 is plotted as a horizontal line.

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.

654.

m=−14m=−14, point (−8,3)(−8,3)

655.

m=35m=35, point (10,6)(10,6)

656.

Horizontal line containing (−2,7)(−2,7)

657.

m=−2m=−2, point (−1,−3)(−1,−3)

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.

658.

(2,10)(2,10) and (−2,−2)(−2,−2)

659.

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

660.

(3,8)(3,8) and (3,−4)(3,−4).

661.

(5,2)(5,2) and (−1,2)(−1,2)

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.

662.

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

663.

line 2x+5y=−102x+5y=−10, point (10,4)(10,4)

664.

line x=4x=4, point (−2,−1)(−2,−1)

665.

line y=−5y=−5, point (−4,3)(−4,3)

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.

666.

line y=−45x+2y=−45x+2, point (8,9)(8,9)

667.

line 2x−3y=92x−3y=9, point (−4,0)(−4,0)

668.

line y=3y=3, point (−1,−3)(−1,−3)

669.

line x=−5x=−5 point (2,1)(2,1)

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.

670.

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

  1. ⓐ (0,1)(0,1)
  2. ⓑ (−2,−4)(−2,−4)
  3. ⓒ (5,2)(5,2)
  4. ⓓ (3,−1)(3,−1)
  5. ⓔ (−1,−5)(−1,−5)
671.

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

  1. ⓐ (6,1)(6,1)
  2. ⓑ (−3,6)(−3,6)
  3. ⓒ (3,2)(3,2)
  4. ⓓ (−5,10)(−5,10)
  5. ⓔ (0,0)(0,0)

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

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

672.

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

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.
673.

Write the inequality shown by the graph with the boundary line y=23x−3y=23x−3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.
674.

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

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 4 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.
675.

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

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus 2 y equals 6 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

Graph Linear Inequalities

In the following exercises, graph each linear inequality.

676.

Graph the linear inequality y>25x−4y>25x−4.

677.

Graph the linear inequality y<−14x+3y<−14x+3.

678.

Graph the linear inequality x−y≤5x−y≤5.

679.

Graph the linear inequality 3x+2y>103x+2y>10.

680.

Graph the linear inequality y≤−3xy≤−3x.

681.

Graph the linear inequality y<6y<6.

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