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4.8: Solve Equations with Fractions

  • Page ID
    114904
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    Learning Objectives

    By the end of this section, you will be able to:

    • Determine whether a fraction is a solution of an equation
    • Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality
    • Solve equations using the Multiplication Property of Equality
    • Translate sentences to equations and solve

    Be Prepared 4.17

    Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

    Evaluate x+4x+4 when x=−3x=−3
    If you missed this problem, review Example 3.23.

    Be Prepared 4.18

    Solve: 2y3=9.2y3=9.
    If you missed this problem, review Example 3.61.

    Be Prepared 4.19

    Solve: y3=−9y3=−9
    If you missed this problem, review Example 4.28.

    Determine Whether a Fraction is a Solution of an Equation

    As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.

    The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction.

    How To

    Determine whether a number is a solution to an equation.

    1. Step 1. Substitute the number for the variable in the equation.
    2. Step 2. Simplify the expressions on both sides of the equation.
    3. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

    Example 4.95

    Determine whether each of the following is a solution of x310=12.x310=12.

    1. x=1x=1
    2. x=45x=45
    3. x=45x=45
    Answer

    .
    . .
    Change to fractions with a LCD of 10. .
    Subtract. .

    Since x=1x=1 does not result in a true equation, 11 is not a solution to the equation.

    .
    . .
    .
    Subtract. .

    Since x=45x=45 results in a true equation, 4545 is a solution to the equation x310=12.x310=12.

    .
    . .
    .
    Subtract. .

    Since x=45x=45 does not result in a true equation, 4545 is not a solution to the equation.

    Try It 4.189

    Determine whether each number is a solution of the given equation.

    x23=16x23=16:

    1. x=1x=1
    2. x=56x=56
    3. x=56x=56

    Try It 4.190

    Determine whether each number is a solution of the given equation.

    y14=38y14=38:

    1. y=1y=1
    2. y=58y=58
    3. y=58y=58

    Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

    In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.

    Addition, Subtraction, and Division Properties of Equality

    For any numbers a,b,a,b, and c,c,

    if a=b,a=b, then a+c=b+c.a+c=b+c. Addition Property of Equality
    if a=b,a=b, then ac=bc.ac=bc. Subtraction Property of Equality
    if a=b,a=b, then ac=bc,c0.ac=bc,c0. Division Property of Equality
    Table 4.3

    In other words, when you add or subtract the same quantity from both sides of an equation, or divide both sides by the same quantity, you still have equality.

    Example 4.96

    Solve: y+916=516.y+916=516.

    Answer

      .
    Subtract 916916 from each side to undo the addition. .
    Simplify on each side of the equation. .
    Simplify the fraction. .
    Check: .
    Substitute y=14y=14. .
    Rewrite as fractions with the LCD. .
    Add. .

    Since y=14y=14 makes y+916=516y+916=516 a true statement, we know we have found the solution to this equation.

    Try It 4.191

    Solve: y+1112=512.y+1112=512.

    Try It 4.192

    Solve: y+815=415.y+815=415.

    We used the Subtraction Property of Equality in Example 4.96. Now we’ll use the Addition Property of Equality.

    Example 4.97

    Solve: a59=89.a59=89.

    Answer

      .
    Add 5959 from each side to undo the subtraction. .
    Simplify on each side of the equation. .
    Simplify the fraction. .
    Check: .
    Substitute a=13a=13. .
    Change to common denominator. .
    Subtract. .

    Since a=13a=13 makes the equation true, we know that a=13a=13 is the solution to the equation.

    Try It 4.193

    Solve: a35=85.a35=85.

    Try It 4.194

    Solve: n37=97.n37=97.

    The next example may not seem to have a fraction, but let’s see what happens when we solve it.

    Example 4.98

    Solve: 10q=44.10q=44.

    Answer

      10q=4410q=44
    Divide both sides by 10 to undo the multiplication. 10q10=441010q10=4410
    Simplify. q=225q=225
    Check:
    Substitute q=225q=225 into the original equation. 10(225)=?4410(225)=?44
    Simplify. 102(225)=?44102(225)=?44
    Multiply. 44=4444=44

    The solution to the equation was the fraction 225.225. We leave it as an improper fraction.

    Try It 4.195

    Solve: 12u=−76.12u=−76.

    Try It 4.196

    Solve: 8m=92.8m=92.

    Solve Equations with Fractions Using the Multiplication Property of Equality

    Consider the equation x4=3.x4=3. We want to know what number divided by 44 gives 3.3. So to “undo” the division, we will need to multiply by 4.4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

    The Multiplication Property of Equality

    For any numbers a,b,a,b, and c,c,

    ifa=b,thenac=bc.ifa=b,thenac=bc.

    If you multiply both sides of an equation by the same quantity, you still have equality.

    Let’s use the Multiplication Property of Equality to solve the equation x7=−9.x7=−9.

    Example 4.99

    Solve: x7=−9.x7=−9.

    Answer

      .
    Use the Multiplication Property of Equality to multiply both sides by 77. This will isolate the variable. .
    Multiply. .
    Simplify. .
    . .
    The equation is true. .

    Try It 4.197

    Solve: f5=−25.f5=−25.

    Try It 4.198

    Solve: h9=−27.h9=−27.

    Example 4.100

    Solve: p−8=−40.p−8=−40.

    Answer

    Here, pp is divided by −8.−8. We must multiply by −8−8 to isolate p.p.

      .
    Multiply both sides by −8−8 .
    Multiply. .
    Simplify. .
    Check:
    Substitute p=320p=320. .
    The equation is true. .

    Try It 4.199

    Solve: c−7=−35.c−7=−35.

    Try It 4.200

    Solve: x−11=−12.x−11=−12.

    Solve Equations with a Coefficient of −1−1

    Look at the equation y=15.y=15. Does it look as if yy is already isolated? But there is a negative sign in front of y,y, so it is not isolated.

    There are three different ways to isolate the variable in this type of equation. We will show all three ways in Example 4.101.

    Example 4.101

    Solve: y=15.y=15.

    Answer

    One way to solve the equation is to rewrite yy as −1y,−1y, and then use the Division Property of Equality to isolate y.y.

    .
    Rewrite yy as −1y−1y. .
    Divide both sides by −1. .
    Simplify each side. .

    Another way to solve this equation is to multiply both sides of the equation by −1.−1.

    .
    Multiply both sides by −1. .
    Simplify each side. .

    The third way to solve the equation is to read yy as “the opposite of yy.” What number has 1515 as its opposite? The opposite of 1515 is −15.−15. So y=−15.y=−15.

    For all three methods, we isolated yy is isolated and solved the equation.

    Check:

    .
    Substitute y=−15y=−15. .
    Simplify. The equation is true. .

    Try It 4.201

    Solve: y=48.y=48.

    Try It 4.202

    Solve: c=−23.c=−23.

    Solve Equations with a Fraction Coefficient

    When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to 1.1.

    For example, in the equation:

    34x=2434x=24

    The coefficient of xExample 4.102.

    Example 4.102

    Solve: 34x=24.34x=24.

    Answer

      .
    Multiply both sides by the reciprocal of the coefficient. .
    Simplify. .
    Multiply. .
    Check: .
    Substitute x=32x=32. .
    Rewrite 3232 as a fraction. .
    Multiply. The equation is true. .

    Notice that in the equation 34x=24,34x=24, we could have divided both sides by 3434 to get xx by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.

    Try It 4.203

    Solve: 25n=14.25n=14.

    Try It 4.204

    Solve: 56y=15.56y=15.

    Example 4.103

    Solve: 38w=72.38w=72.

    Answer

    The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.

      .
    Multiply both sides by the reciprocal of 3838. .
    Simplify; reciprocals multiply to one. .
    Multiply. .
    Check: .
    Let w=−192w=−192. .
    Multiply. It checks. .

    Try It 4.205

    Solve: 47a=52.47a=52.

    Try It 4.206

    Solve: 79w=84.79w=84.

    Translate Sentences to Equations and Solve

    Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

    Subtraction Property of Equality:
    For any real numbers a, b,a, b, and c,c,

    if a=b,a=b, then ac=bc.ac=bc.
    Addition Property of Equality:
    For any real numbers a, b,a, b, and c,c,

    if a=b,a=b, then a+c=b+c.a+c=b+c.
    Division Property of Equality:
    For any numbers a, b,a, b, and c,c, where c0c0

    if a=b,a=b, then ac=bcac=bc
    Multiplication Property of Equality:
    For any real numbers a, b,a, b, and cc

    if a=b,a=b, then ac=bcac=bc

    When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

    In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions.

    Example 4.104

    Translate and solve: nn divided by 66 is −24.−24.

    Answer

    Translate. .
    Multiply both sides by 66. .
    Simplify. .
    Check: Is −144−144 divided by 66 equal to −24−24?
    Translate. .
    Simplify. It checks. .

    Try It 4.207

    Translate and solve: nn divided by 77 is equal to −21.−21.

    Try It 4.208

    Translate and solve: nn divided by 88 is equal to −56.−56.

    Example 4.105

    Translate and solve: The quotient of qq and −5−5 is 70.70.

    Answer

    Translate. .
    Multiply both sides by −5−5. .
    Simplify. .
    Check: Is the quotient of −350−350 and −5−5 equal to 7070?
    Translate. .
    Simplify. It checks. .

    Try It 4.209

    Translate and solve: The quotient of qq and −8−8 is 72.72.

    Try It 4.210

    Translate and solve: The quotient of pp and −9−9 is 81.81.

    Example 4.106

    Translate and solve: Two-thirds of ff is 18.18.

    Answer

    Translate. .
    Multiply both sides by 3232. .
    Simplify. .
    Check: Is two-thirds of 2727 equal to 1818?
    Translate. .
    Simplify. It checks. .

    Try It 4.211

    Translate and solve: Two-fifths of ff is 16.16.

    Try It 4.212

    Translate and solve: Three-fourths of ff is 21.21.

    Example 4.107

    Translate and solve: The quotient of mm and 5656 is 34.34.

    Answer

      The quotient of mm and 5656 is 3434.
    Translate. m56=34m56=34
    Multiply both sides by 5656 to isolate mm. 56(m56)=56(34)56(m56)=56(34)
    Simplify. m=5·36·4m=5·36·4
    Remove common factors and multiply. m=58m=58
    Check:
    Is the quotient of 5858 and 5656 equal to 3434? 5856=?345856=?34
    Rewrite as division. 58÷56=?3458÷56=?34
    Multiply the first fraction by the reciprocal of the second. 58·65=?3458·65=?34
    Simplify. 34=3434=34

    Our solution checks.

    Try It 4.213

    Translate and solve. The quotient of nn and 2323 is 512.512.

    Try It 4.214

    Translate and solve The quotient of cc and 3838 is 49.49.

    Example 4.108

    Translate and solve: The sum of three-eighths and xx is three and one-half.

    Answer

    Translate. .
    Use the Subtraction Property of Equality to subtract 3838 from both sides. .
    Combine like terms on the left side. .
    Convert mixed number to improper fraction. .
    Convert to equivalent fractions with LCD of 8. .
    Subtract. .
    Write as a mixed number. .

    We write the answer as a mixed number because the original problem used a mixed number.

    Check:

    Is the sum of three-eighths and 318318 equal to three and one-half?

    38+318=?31238+318=?312
    Add. 348=?312348=?312
    Simplify. 312=312312=312

    The solution checks.

    Try It 4.215

    Translate and solve: The sum of five-eighths and xx is one-fourth.

    Try It 4.216

    Translate and solve: The difference of one-and-three-fourths and xx is five-sixths.

    Media

    Section 4.7 Exercises

    Practice Makes Perfect

    Determine Whether a Fraction is a Solution of an Equation

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

    498.

    x25=110x25=110:

    1. x=1x=1
    2. x=12x=12
    3. x=12x=12
    499.

    y13=512y13=512:

    1. y=1y=1
    2. y=34y=34
    3. y=34y=34
    500.

    h+34=25h+34=25:

    1. h=1h=1
    2. h=720h=720
    3. h=720h=720
    501.

    k+25=56k+25=56:

    1. k=1k=1
    2. k=1330k=1330
    3. k=1330k=1330

    Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

    In the following exercises, solve.

    502.

    y + 1 3 = 4 3 y + 1 3 = 4 3

    503.

    m + 3 8 = 7 8 m + 3 8 = 7 8

    504.

    f + 9 10 = 2 5 f + 9 10 = 2 5

    505.

    h + 5 6 = 1 6 h + 5 6 = 1 6

    506.

    a 5 8 = 7 8 a 5 8 = 7 8

    507.

    c 1 4 = 5 4 c 1 4 = 5 4

    508.

    x ( 3 20 ) = 11 20 x ( 3 20 ) = 11 20

    509.

    z ( 5 12 ) = 7 12 z ( 5 12 ) = 7 12

    510.

    n 1 6 = 3 4 n 1 6 = 3 4

    511.

    p 3 10 = 5 8 p 3 10 = 5 8

    512.

    s + ( 1 2 ) = 8 9 s + ( 1 2 ) = 8 9

    513.

    k + ( 1 3 ) = 4 5 k + ( 1 3 ) = 4 5

    514.

    5 j = 17 5 j = 17

    515.

    7 k = 18 7 k = 18

    516.

    −4 w = 26 −4 w = 26

    517.

    −9 v = 33 −9 v = 33

    Solve Equations with Fractions Using the Multiplication Property of Equality

    In the following exercises, solve.

    518.

    f 4 = −20 f 4 = −20

    519.

    b 3 = −9 b 3 = −9

    520.

    y 7 = −21 y 7 = −21

    521.

    x 8 = −32 x 8 = −32

    522.

    p −5 = −40 p −5 = −40

    523.

    q −4 = −40 q −4 = −40

    524.

    r −12 = −6 r −12 = −6

    525.

    s −15 = −3 s −15 = −3

    526.

    x = 23 x = 23

    527.

    y = 42 y = 42

    528.

    h = 5 12 h = 5 12

    529.

    k = 17 20 k = 17 20

    530.

    4 5 n = 20 4 5 n = 20

    531.

    3 10 p = 30 3 10 p = 30

    532.

    3 8 q = −48 3 8 q = −48

    533.

    5 2 m = −40 5 2 m = −40

    534.

    2 9 a = 16 2 9 a = 16

    535.

    3 7 b = 9 3 7 b = 9

    536.

    6 11 u = −24 6 11 u = −24

    537.

    5 12 v = −15 5 12 v = −15

    Mixed Practice

    In the following exercises, solve.

    538.

    3 x = 0 3 x = 0

    539.

    8 y = 0 8 y = 0

    540.

    4 f = 4 5 4 f = 4 5

    541.

    7 g = 7 9 7 g = 7 9

    542.

    p + 2 3 = 1 12 p + 2 3 = 1 12

    543.

    q + 5 6 = 1 12 q + 5 6 = 1 12

    544.

    7 8 m = 1 10 7 8 m = 1 10

    545.

    1 4 n = 7 10 1 4 n = 7 10

    546.

    2 5 = x + 3 4 2 5 = x + 3 4

    547.

    2 3 = y + 3 8 2 3 = y + 3 8

    548.

    11 20 = - f 11 20 = - f

    549.

    8 15 = - d 8 15 = - d

    Translate Sentences to Equations and Solve

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

    550.

    nn divided by eight is −16.−16.

    551.

    nn divided by six is −24.−24.

    552.

    mm divided by −9−9 is −7.−7.

    553.

    mm divided by −7−7 is −8.−8.

    554.

    The quotient of ff and −3−3 is −18.−18.

    555.

    The quotient of ff and −4−4 is −20.−20.

    556.

    The quotient of gg and twelve is 8.8.

    557.

    The quotient of gg and nine is 14.14.

    558.

    Three-fourths of qq is 12.12.

    559.

    Two-fifths of qq is 20.20.

    560.

    Seven-tenths of pp is −63.−63.

    561.

    Four-ninths of pp is −28.−28.

    562.

    mm divided by 44 equals negative 6.6.

    563.

    The quotient of hh and 22 is 43.43.

    564.

    Three-fourths of zz is 15.15.

    565.

    The quotient of aa and 2323 is 34.34.

    566.

    The sum of five-sixths and xx is 12.12.

    567.

    The sum of three-fourths and xx is 18.18.

    568.

    The difference of yy and one-fourth is 18.18.

    569.

    The difference of yy and one-third is 16.16.

    Everyday Math

    570.

    Shopping Teresa bought a pair of shoes on sale for $48$48. The sale price was 2323 of the regular price. Find the regular price of the shoes by solving the equation 23p=4823p=48

    571.

    Playhouse The table in a child’s playhouse is 3535 of an adult-size table. The playhouse table is 1818 inches high. Find the height of an adult-size table by solving the equation 35h=18.35h=18.

    Writing Exercises

    572.

    Example 4.100 describes three methods to solve the equation y=15.y=15. Which method do you prefer? Why?

    573.

    Richard thinks the solution to the equation 34x=2434x=24 is 16.16. Explain why Richard is wrong.

    Self Check

    After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    .

    Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


    This page titled 4.8: Solve Equations with Fractions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

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