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2.3.6: Two-Step Equations with Subtraction and Division

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    Solve Equations Involving Inverse Properties of Subtraction and Division

    Figure 2.3.6.1

    Brandon and Felicia sold rolls of wrapping paper for a school fundraiser. Brandon sold 3 less than half the number of rolls that Felicia sold. Brandon sold a total of 9 rolls of wrapping paper.

    Write an algebraic equation to represent w, Remember that w is the number of rolls of wrap the number of rolls of wrapping paper that Felicia sold. Then, find the number of rolls of wrapping paper that Felicia sold.

    In this concept, you will solve equations involving inverse properties of subtraction and division.

    Inverse Properties of Subtracting and Dividing

    To solve a two-step equation, you will need to use more than one inverse operation. You begin solving two-step equations by isolating the variable.

    For example, solve for z.

    z6−7=3

    Your first step should be to use inverse operations to get the term that includes a variable, z6, by itself on one side of the equal sign. In the equation, 7 is subtracted from z6. So, you can use the inverse of subtraction—addition. Therefore, add 7 to both sides of the equation.

    z6−7z6−7+7z6===33+710

    Next, you can use inverse operations to isolate z. Since z is divided by 6, you can use the inverse of division – multiplication.

    z6z6×6z===1010×660

    The answer is 60.

    Let’s review your steps to solving this two-step equation.

    Figure 2.3.6.2

    Examples

    Example 2.3.6.1

    Earlier, you were given a problem about Brandon and Felicia’s fundraising.

    Brandon sold 3 less than half of Felicia’s total, which was 9 rolls of wrap.

    Solution

    First, use the key words from the question to help you translate the problem into an equation. Remember that w is the number of rolls of wrap.

    Brand on sold 3 less than half the number that Felicia sold.↓↓ −3 w2

    So the equation is w2−3=9.

    Next, use the inverse of subtraction—addition to isolate the variable.

    (w/2)−3=9

    (w/2)−3+3=9+3

    w/2=12

    Then, you can use inverse operations to isolate w. Since w is divided by 2, you can use the inverse of division – multiplication.

    w/2=12

    (w/2)×2=12x2

    w=24

    The answer is 24.

    Felicia sold 24 rolls of wrapping paper.

    Example 2.3.6.2

    (x/6)−9=8

    Solution

    First, use the inverse of subtraction—addition to isolate the variable.

    (x/6)−9=8

    (x/6)−9+9=8+9

    (x/6)=17

    Next, you can use inverse operations to isolate x. Since x is divided by 6, you can use the inverse of division – multiplication.

    x/6=17

    (x/6)×6=17x6

    x=102

    The answer is 102.

    Example 2.3.6.3

    (x/3)−8=9

    Solution

    First, use the inverse of subtraction—addition to isolate the variable.

    (x/3)−8=9

    (x/3)−8+8=9+8

    x/3=17

    Next, you can use inverse operations to isolate x. Since x is divided by 3, you can use the inverse of division – multiplication.

    x/3=17

    (x/3)×3=17x3

    x=51

    The answer is 51.

    Example 2.3.6.4

    (y/7)−2=13

    Solution

    First, use the inverse of subtraction—addition to isolate the variable.

    (y/7)−2=13

    (y/7)−2+2=13+2

    y/7=15

    Next, you can use inverse operations to isolate y. Since y is divided by 73, you can use the inverse of division – multiplication.

    y/7=15

    (y/7)×7=15x7

    y=105

    The answer is 105.

    Example 2.3.6.5

    (a/7)−2=12

    Solution

    First, use the inverse of subtraction—addition to isolate the variable.

    (a/7)−2=12

    (a/7)−2+2=12+2

    a/7=14

    Next, you can use inverse operations to isolate a. Since a is divided by 7, you can use the inverse of division – multiplication.

    a7=14

    a7×7=14x7

    a=98

    The answer is 98.

    Review

    Solve each two-step equation that has division and subtraction in it.

    1. x/5−4=8
    2. y/6−3=8
    3. x/7−7=10
    4. x/8−4=12
    5. y/7−5=11
    6. x/4−10=12
    7. y/4−8=2
    8. x/3−12=9
    9. a/5−3=11
    10. b/4−1=15
    11. x/2−8=4
    12. a/7−4=9
    13. b/4−7=3
    14. x/8−1=12
    15. y/6−8=5
    16. x/2−15=12

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 3.4.

    Vocabulary

    Term Definition
    Algebraic Equation An algebraic equation contains numbers, variables, operations, and an equals sign.
    Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
    inverse Inverse functions are functions that 'undo' each other. Formally: f(x) and g(x) are inverse functions if f(g(x))=g(f(x))=x.
    One-Step Equation A one-step equation is an algebraic equation with one operation in it that requires one step to solve.
    Two-Step Equation A two-step equation is an algebraic equation with two operations in it that requires two steps to solve.
    Variable A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

    Additional Resources

    Video:

    Practice: Two-Step Equations with Subtraction and Division

    This page titled 2.3.6: Two-Step Equations with Subtraction and Division is shared under a CC BY-NC license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform.