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8.E: Solving Linear Equations (Exercises)

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8.1 - Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, determine whether the given number is a solution to the equation.

• x + 16 = 31, x = 15
• w − 8 = 5, w = 3
• −9n = 45, n = 54
• 4a = 72, a = 18

In the following exercises, solve the equation using the Subtraction Property of Equality.

• y + 2 = −6
• a + \(\dfrac{1}{3} = \dfrac{5}{3}\)
• n + 3.6 = 5.1

In the following exercises, solve the equation using the Addition Property of Equality.

• u − 7 = 10
• x − 9 = −4
• c − \(\dfrac{3}{11} = \dfrac{9}{11}\)
• p − 4.8 = 14

In the following exercises, solve the equation.

• n − 12 = 32
• y + 16 = −9
• f + \(\dfrac{2}{3}\) = 4
• d − 3.9 = 8.2
• y + 8 − 15 = −3
• 7x + 10 − 6x + 3 = 5
• 6(n − 1) − 5n = −14
• 8(3p + 5) − 23(p − 1) = 35

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

• The sum of −6 and m is 25.
• Four less than n is 13.

In the following exercises, translate into an algebraic equation and solve.

• Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?
• Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?
• Peter paid $9.75 to go to the movies, which was $46.25 less than he paid to go to a concert. How much did he pay for the concert?
• Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week?

8.2 - Solve Equations using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division Property of Equality.

• 13a = −65
• 0.25p = 5.25
• −y = 4

In the following exercises, solve each equation using the Multiplication Property of Equality.

• \(\dfrac{n}{6}\) = 18
• y −10 = 30
• 36 = \(\dfrac{3}{4}\)x
• \(\dfrac{5}{8} u = \dfrac{15}{16}\)

In the following exercises, solve each equation.

• −18m = −72
• \(\dfrac{c}{9}\) = 36
• 0.45x = 6.75
• \(\dfrac{11}{12} = \dfrac{2}{3} y\)
• 5r − 3r + 9r = 35 − 2
• 24x + 8x − 11x = −7−14

8.3 - Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the equations with constants on both sides.

• 8p + 7 = 47
• 10w − 5 = 65
• 3x + 19 = −47
• 32 = −4 − 9n

In the following exercises, solve the equations with variables on both sides.

• 7y = 6y − 13
• 5a + 21 = 2a
• k = −6k − 35
• 4x − \(\dfrac{3}{8}\) = 3x

In the following exercises, solve the equations with constants and variables on both sides.

• 12x − 9 = 3x + 45
• 5n − 20 = −7n − 80
• 4u + 16 = −19 − u
• \(\dfrac{5}{8} c\) − 4 = \(\dfrac{3}{8} c\) + 4

In the following exercises, solve each linear equation using the general strategy.

• 6(x + 6) = 24
• 9(2p − 5) = 72
• −(s + 4) = 18
• 8 + 3(n − 9) = 17
• 23 − 3(y − 7) = 8
• \(\dfrac{1}{3}\)(6m + 21) = m − 7
• 8(r − 2) = 6(r + 10)
• 5 + 7(2 − 5x) = 2(9x + 1) − (13x − 57)
• 4(3.5y + 0.25) = 365
• 0.25(q − 8) = 0.1(q + 7)

8.4 - Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation by clearing the fractions.

• \(\dfrac{2}{5} n − \dfrac{1}{10} = \dfrac{7}{10}\)
• \(\dfrac{1}{3} x + \dfrac{1}{5} x = 8\)
• \(\dfrac{3}{4} a − \dfrac{1}{3} = \dfrac{1}{2} a + \dfrac{5}{6}\)
• \(\dfrac{1}{2}\)(k + 3) = \(\dfrac{1}{3}\)(k + 16)

In the following exercises, solve each equation by clearing the decimals.

• 0.8x − 0.3 = 0.7x + 0.2
• 0.36u + 2.55 = 0.41u + 6.8
• 0.6p − 1.9 = 0.78p + 1.7
• 0.10d + 0.05(d − 4) = 2.05

PRACTICE TEST

• \(\dfrac{23}{5}\)
• n − 18 = 31
• 4y − 8 = 16
• −8x − 15 + 9x − 1 = −21
• −15a = 120
• \(\dfrac{2}{3}\)x = 6
• x + 3.8 = 8.2
• 10y = −5y + 60
• 8n + 2 = 6n + 12
• 9m − 2 − 4m + m = 42 − 8
• −5(2x + 1) = 45
• −(d + 9) = 23
• 2(6x + 5) − 8 = −22
• 8(3a + 5) − 7(4a − 3) = 20 − 3a
• \(\dfrac{1}{4} p + \dfrac{1}{3} = \dfrac{1}{2}\)
• 0.1d + 0.25(d + 8) = 4.1
• Translate and solve: The difference of twice x and 4 is 16.
• Samuel paid $25.82 for gas this week, which was $3.47 less than he paid last week. How much did he pay last week?

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/[email protected] ."

Practice Test

For the following exercises, determine whether each of the following relations is a function.

y = 2 x + 8 y = 2 x + 8

{ ( 2 , 1 ) , ( 3 , 2 ) , ( − 1 , 1 ) , ( 0 , − 2 ) } { ( 2 , 1 ) , ( 3 , 2 ) , ( − 1 , 1 ) , ( 0 , − 2 ) }

For the following exercises, evaluate the function f ( x ) = − 3 x 2 + 2 x f ( x ) = − 3 x 2 + 2 x at the given input.

f ( −2 ) f ( −2 )

f ( a ) f ( a )

Show that the function f ( x ) = − 2 ( x − 1 ) 2 + 3 f ( x ) = − 2 ( x − 1 ) 2 + 3 is not one-to-one.

Write the domain of the function f ( x ) = 3 − x f ( x ) = 3 − x in interval notation.

Given f ( x ) = 2 x 2 − 5 x , f ( x ) = 2 x 2 − 5 x , find f ( a + 1 ) − f ( 1 ) f ( a + 1 ) − f ( 1 ) in simplest form.

Graph the function f ( x ) = { x + 1   if − 2 < x < 3    − x    if   x ≥ 3 f ( x ) = { x + 1   if − 2 < x < 3    − x    if   x ≥ 3

Find the average rate of change of the function f ( x ) = 3 − 2 x 2 + x f ( x ) = 3 − 2 x 2 + x by finding f ( b ) − f ( a ) b − a f ( b ) − f ( a ) b − a in simplest form.

For the following exercises, use the functions f ( x ) = 3 − 2 x 2 + x and  g ( x ) = x f ( x ) = 3 − 2 x 2 + x and  g ( x ) = x to find the composite functions.

( g ∘ f ) ( x ) ( g ∘ f ) ( x )

( g ∘ f ) ( 1 ) ( g ∘ f ) ( 1 )

Express H ( x ) = 5 x 2 − 3 x 3 H ( x ) = 5 x 2 − 3 x 3 as a composition of two functions, f f and g , g , where ( f ∘ g ) ( x ) = H ( x ) . ( f ∘ g ) ( x ) = H ( x ) .

For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

f ( x ) = x + 6 − 1 f ( x ) = x + 6 − 1

f ( x ) = 1 x + 2 − 1 f ( x ) = 1 x + 2 − 1

For the following exercises, determine whether the functions are even, odd, or neither.

f ( x ) = − 5 x 2 + 9 x 6 f ( x ) = − 5 x 2 + 9 x 6

f ( x ) = − 5 x 3 + 9 x 5 f ( x ) = − 5 x 3 + 9 x 5

f ( x ) = 1 x f ( x ) = 1 x

Graph the absolute value function f ( x ) = − 2 | x − 1 | + 3. f ( x ) = − 2 | x − 1 | + 3.

For the following exercises, find the inverse of the function.

f ( x ) = 3 x − 5 f ( x ) = 3 x − 5

f ( x ) = 4 x + 7 f ( x ) = 4 x + 7

For the following exercises, use the graph of g g shown in Figure 1 .

On what intervals is the function increasing?

On what intervals is the function decreasing?

Approximate the local minimum of the function. Express the answer as an ordered pair.

Approximate the local maximum of the function. Express the answer as an ordered pair.

For the following exercises, use the graph of the piecewise function shown in Figure 2 .

Find f ( 2 ) . f ( 2 ) .

Find f ( −2 ) . f ( −2 ) .

Write an equation for the piecewise function.

For the following exercises, use the values listed in Table 1 .

Find F ( 6 ) . F ( 6 ) .

Solve the equation F ( x ) = 5. F ( x ) = 5.

Is the graph increasing or decreasing on its domain?

Is the function represented by the graph one-to-one?

Find F − 1 ( 15 ) . F − 1 ( 15 ) .

Given f ( x ) = − 2 x + 11 , f ( x ) = − 2 x + 11 , find f − 1 ( x ) . f − 1 ( x ) .

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Access for free at https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra
• Publication date: Feb 13, 2015
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra/pages/3-practice-test

© Dec 8, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

8th Grade Linear Equations Worksheets

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