7.4: Distributive Property (2024)

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    Learning Objectives
    • Simplify expressions using the distributive property
    • Evaluate expressions using the distributive property
    be prepared!

    Before you get started, take this readiness quiz.

    1. Multiply: 3(0.25). If you missed this problem, review Example 5.3.5
    2. Simplify: 10 − (−2)(3). If you missed this problem, review Example 3.7.5.
    3. Combine like terms: 9y + 17 + 3y − 2. If you missed this problem, review Example 2.3.10.

    Simplify Expressions Using the Distributive Property

    Suppose three friends are going to the movies. They each need $9.25; that is, 9 dollars and 1 quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

    7.4: Distributive Property (2)

    7.4: Distributive Property (3)

    They need 3 times $9, so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the Distributive Property.

    Definition: Distributive Property

    If a, b, c are real numbers, then a(b + c) = ab + ac.

    Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

    \[\begin{split} 3(9&.25) \\ 3(9 &+ 0.25) \\ 3(9) &+ 3(0.25) \\ 27 &+ 0.75 \\ 27&.75 \end{split}\]

    In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression 3(x + 4), the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the Distributive Property, as shown in Example \(\PageIndex{1}\).

    Example \(\PageIndex{1}\):

    Simplify: 3(x + 4).

    Solution

    Distribute. 3 • x + 3 • 4
    Multiply. 3x + 12
    Exercise \(\PageIndex{1}\):

    Simplify: 4(x + 2).

    Answer

    \(4x+8\)

    Exercise \(\PageIndex{2}\):

    Simplify: 6(x + 7).

    Answer

    6x + 42

    Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example 7.17 would look like this:

    7.4: Distributive Property (4)

    \[3 \cdot x + 3 \cdot 4\]

    Example \(\PageIndex{2}\):

    Simplify: 6(5y + 1).

    Solution

    7.4: Distributive Property (5)

    Distribute. $$6 \cdot 5y + 6 \cdot 1$$
    Multiply. $$30y + 6$$
    Exercise \(\PageIndex{3}\):

    Simplify: 9(3y + 8).

    Answer

    27y + 72

    Exercise \(\PageIndex{4}\):

    Simplify: 5(5w + 9).

    Answer

    25w + 45

    The distributive property can be used to simplify expressions that look slightly different from a(b + c). Here are two other forms.

    Definition: Distributive Property

    If a, b, c are real numbers, then\[a(b + c) = ab + ac$$Other forms$$a(b − c) = ab − ac$$$$(b + c)a = ba + ca\]

    Example \(\PageIndex{3}\):

    Simplify: 2(x − 3).

    Solution

    7.4: Distributive Property (6)

    Distribute. $$2 \cdot x + 2 \cdot 3$$
    Multiply. $$2x - 6$$
    Exercise \(\PageIndex{5}\):

    Simplify: 7(x − 6).

    Answer

    7x - 42

    Exercise \(\PageIndex{6}\):

    Simplify: 8(x − 5).

    Answer

    8x - 40

    Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

    Example \(\PageIndex{4}\):

    Simplify: \(\dfrac{3}{4}\)(n + 12).

    Solution

    7.4: Distributive Property (7)

    Distribute. $$\dfrac{3}{4} \cdot n + \dfrac{3}{4} \cdot 12$$
    Multiply. $$\dfrac{3}{4}n + 9$$
    Exercise \(\PageIndex{7}\):

    Simplify: \(\dfrac{2}{5}\)(p + 10).

    Answer

    \(\frac{2}{5}p + 4 \)

    Exercise \(\PageIndex{8}\):

    Simplify: \(\dfrac{3}{7}\)(u + 21).

    Answer

    \(\frac{3}{7}u +9 \)

    Example \(\PageIndex{5}\):

    Simplify: \(8 \left(\dfrac{3}{8}x + \dfrac{1}{4}\right)\).

    Solution

    7.4: Distributive Property (8)

    Distribute. $$8 \cdot \dfrac{3}{8}x + 8 \cdot \dfrac{1}{4}$$
    Multiply. $$3x + 2$$
    Exercise \(\PageIndex{9}\):

    Simplify: \(6 \left(\dfrac{5}{6}y + \dfrac{1}{2}\right)\).

    Answer

    5y + 3

    Exercise \(\PageIndex{10}\):

    Simplify: \(12 \left(\dfrac{1}{3}n + \dfrac{3}{4}\right)\).

    Answer

    4n + 9

    Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

    Example \(\PageIndex{6}\):

    Simplify: 100(0.3 + 0.25q).

    Solution

    7.4: Distributive Property (9)

    Distribute. $$100(0.3) + 100(0.25q)$$
    Multiply. $$30 + 25q$$
    Exercise \(\PageIndex{11}\):

    Simplify: 100(0.7 + 0.15p).

    Answer

    70 + 15p

    Exercise \(\PageIndex{12}\):

    Simplify: 100(0.04 + 0.35d).

    Answer

    4 + 35d

    In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

    Example \(\PageIndex{7}\):

    Simplify: \(m(n − 4)\).

    Solution

    7.4: Distributive Property (10)

    Distribute. $$m \cdot n - m \cdot 4$$
    Multiply. $$mn - 4m$$

    Notice that we wrote m • 4 as 4m. We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.

    Exercise \(\PageIndex{13}\):

    Simplify: r(s − 2).

    Answer

    rs - 2r

    Exercise \(\PageIndex{14}\):

    Simplify: y(z − 8).

    Answer

    yz - 8y

    The next example will use the ‘backwards’ form of the Distributive Property, (b + c)a = ba + ca.

    Example \(\PageIndex{8}\):

    Simplify: (x + 8)p.

    Solution

    7.4: Distributive Property (11)

    Distribute. $$px + 8p$$
    Exercise \(\PageIndex{15}\):

    Simplify: (x + 2)p.

    Answer

    xp + 2p

    Exercise \(\PageIndex{16}\):

    Simplify: (y + 4)q.

    Answer

    yq + 4q

    When you distribute a negative number, you need to be extra careful to get the signs correct.

    Example \(\PageIndex{9}\):

    Simplify: −2(4y + 1).

    Solution

    7.4: Distributive Property (12)

    Distribute. $$-2 \cdot 4y + (-2) \cdot 1$$
    Simplify. $$-8y - 2$$
    Exercise \(\PageIndex{17}\):

    Simplify: −3(6m + 5).

    Answer

    -18m - 15

    Exercise \(\PageIndex{18}\):

    Simplify: −6(8n + 11).

    Answer

    -48n - 66

    Example \(\PageIndex{10}\):

    Simplify: −11(4 − 3a).

    Solution

    Distribute. $$-11 \cdot 4 - (-11) \cdot 3a$$
    Multiply. $$-44 - (-33a)$$
    Simplify. $$-44 + 33a$$

    You could also write the result as 33a − 44. Do you know why?

    Exercise \(\PageIndex{19}\):

    Simplify: −5(2 − 3a).

    Answer

    -10 + 15a

    Exercise \(\PageIndex{20}\):

    Simplify: −7(8 − 15y).

    Answer

    -56 + 105y

    In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, −a = −1 • a.

    Example \(\PageIndex{11}\):

    Simplify: −(y + 5).

    Solution

    Multiplying by −1 results in the opposite. $$-1 (y + 5)$$
    Distribute. $$-1 \cdot y + (-1) \cdot 5$$
    Simplify. $$-y + (-5)$$
    Simplify. $$-y -5$$
    Exercise \(\PageIndex{21}\):

    Simplify: −(z − 11).

    Answer

    -z + 11

    Exercise \(\PageIndex{22}\):

    Simplify: −(x − 4).

    Answer

    -x + 4

    Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

    Example \(\PageIndex{12}\):

    Simplify: 8 − 2(x + 3).

    Solution

    Distribute. $$8 - 2 \cdot x - 2 \cdot 3$$
    Multiply. $$8 - 2x - 6$$
    Combine like terms. $$-2x + 2$$
    Exercise \(\PageIndex{23}\):

    Simplify: 9 − 3(x + 2).

    Answer

    -3x + 3

    Exercise \(\PageIndex{24}\):

    Simplify: 7x − 5(x + 4).

    Answer

    2x - 20

    Example \(\PageIndex{13}\):

    Simplify: 4(x − 8) − (x + 3).

    Solution

    Distribute. $$4x - 32 - x - 3$$
    Combine like terms. $$3x - 35$$
    Exercise \(\PageIndex{25}\):

    Simplify: 6(x − 9) − (x + 12).

    Answer

    5x - 66

    Exercise \(\PageIndex{26}\):

    Simplify: 8(x − 1) − (x + 5).

    Answer

    7x - 13

    Evaluate Expressions Using the Distributive Property

    Some students need to be convinced that the Distributive Property always works. In the examples below, we will practice evaluating some of the expressions from previous examples; in part (a), we will evaluate the form with parentheses, and in part (b) we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

    Example \(\PageIndex{14}\):

    When y = 10 evaluate: (a) 6(5y + 1) (b) 6 • 5y + 6 • 1.

    Solution

    (a) 6(5y + 1)

    Substitute \(\textcolor{red}{10}\) for y. $$6(5 \cdot \textcolor{red}{10} + 1)$$
    Simplify in the parentheses. $$6(51)$$
    Multiply. $$306$$

    (b) 6 • 5y + 6 • 1

    Substitute \(\textcolor{red}{10}\) for y. $$6 \cdot 5 \cdot \textcolor{red}{10} + 6 \cdot 1$$
    Simplify. $$300 + 6$$
    Add. $$306$$

    Notice, the answers are the same. When y = 10, 6(5y + 1) = 6 • 5y + 6 • 1. Try it yourself for a different value of y.

    Exercise \(\PageIndex{27}\):

    Evaluate when w = 3: (a) 5(5w + 9) (b) 5 • 5w + 5 • 9.

    Answer a

    \(120\)

    Answer b

    \(120\)

    Exercise \(\PageIndex{28}\):

    Evaluate when y = 2: (a) 9(3y + 8) (b) 9 • 3y + 9 • 8.

    Answer a

    \(126\)

    Answer b

    \(126\)

    Example \(\PageIndex{15}\):

    When y = 3, evaluate (a) −2(4y + 1) (b) −2 • 4y + (−2) • 1.

    Solution

    (a) −2(4y + 1)

    Substitute \(\textcolor{red}{3}\) for y. $$-2(4 \cdot \textcolor{red}{3} + 1)$$
    Simplify in the parentheses. $$-2(13)$$
    Multiply. $$-26$$

    (b) −2 • 4y + (−2) • 1

    Substitute \(\textcolor{red}{3}\) for y. $$-2 \cdot 4 \cdot \textcolor{red}{3} + (-2) \cdot 1$$
    Multiply. $$-24 - 2$$
    Subtract. $$-26$$
    The answers are the same when y = 3. $$-2(4y + 1) = -8y - 2$$
    Exercise \(\PageIndex{29}\):

    Evaluate when n = −2: (a) −6(8n + 11) (b) −6 • 8n + (−6) • 11.

    Answer a

    \(30\)

    Answer b

    \(30\)

    Exercise \(\PageIndex{30}\):

    Evaluate when m = −1: (a) −3(6m + 5) (b) −3 • 6m + (−3) • 5.

    Answer a

    \(3\)

    Answer b

    \(3\)

    Example \(\PageIndex{16}\):

    When y = 35 evaluate (a) −(y + 5) and (b) −y − 5 to show that −(y + 5) = −y − 5.

    Solution

    (a) −(y + 5)

    Substitute \(\textcolor{red}{35}\) for y. $$-(\textcolor{red}{35} + 5)$$
    Add in the parentheses. $$-(40)$$
    Simplify. $$-40$$

    (b) −y − 5

    Substitute \(\textcolor{red}{35}\) for y. $$-\textcolor{red}{35} - 5$$
    Simplify. $$-40$$
    The answers are the same when y = 35, demonstrating that $$-(y + 5) = -y - 5$$
    Exercise \(\PageIndex{31}\):

    Evaluate when x = 36: (a) −(x − 4) (b) −x + 4 to show that −(x − 4) = − x + 4.

    Answer a

    \(-32\)

    Answer b

    \(-32\)

    Exercise \(\PageIndex{32}\):

    Evaluate when z = 55: (a) −(z − 10) (b) −z + 10 to show that −(z − 10) = − z + 10.

    Answer a

    \(-45\)

    Answer b

    \(-45\)

    ACCESS ADDITIONAL ONLINE RESOURCES

    Model Distribution

    The Distributive Property

    Practice Makes Perfect

    Simplify Expressions Using the Distributive Property

    In the following exercises, simplify using the distributive property.

    1. 4(x + 8)
    2. 3(a + 9)
    3. 8(4y + 9)
    4. 9(3w + 7)
    5. 6(c − 13)
    6. 7(y − 13)
    7. 7(3p − 8)
    8. 5(7u − 4)
    9. \(\dfrac{1}{2}\)(n + 8)
    10. \(\dfrac{1}{3}\)(u + 9)
    11. \(\dfrac{1}{4}\)(3q + 12)
    12. \(\dfrac{1}{5}\)(4m + 20)
    13. \(9 \left(\dfrac{5}{9} y − \dfrac{1}{3}\right)\)
    14. \(10 \left(\dfrac{3}{10} x − \dfrac{2}{5}\right)\)
    15. \(12 \left(\dfrac{1}{4} + \dfrac{2}{3} r\right)\)
    16. \(12 \left(\dfrac{1}{6} + \dfrac{3}{4} s\right)\)
    17. r(s − 18)
    18. u(v − 10)
    19. (y + 4)p
    20. (a + 7)x
    21. −2(y + 13)
    22. −3(a + 11)
    23. −7(4p + 1)
    24. −9(9a + 4)
    25. −3(x − 6)
    26. −4(q − 7)
    27. −9(3a − 7)
    28. −6(7x − 8)
    29. −(r + 7)
    30. −(q + 11)
    31. −(3x − 7)
    32. −(5p − 4)
    33. 5 + 9(n − 6)
    34. 12 + 8(u − 1)
    35. 16 − 3(y + 8)
    36. 18 − 4(x + 2)
    37. 4 − 11(3c − 2)
    38. 9 − 6(7n − 5)
    39. 22 − (a + 3)
    40. 8 − (r − 7)
    41. −12 − (u + 10)
    42. −4 − (c − 10)
    43. (5m − 3) − (m + 7)
    44. (4y − 1) − (y − 2)
    45. 5(2n + 9) + 12(n − 3)
    46. 9(5u + 8) + 2(u − 6)
    47. 9(8x − 3) − (−2)
    48. 4(6x − 1) − (−8)
    49. 14(c − 1) − 8(c − 6)
    50. 11(n − 7) − 5(n − 1)
    51. 6(7y + 8) − (30y − 15)
    52. 7(3n + 9) − (4n − 13)

    Evaluate Expressions Using the Distributive Property

    In the following exercises, evaluate both expressions for the given value.

    1. If v = −2, evaluate
      1. 6(4v + 7)
      2. 6 · 4v + 6 · 7
    2. If u = −1, evaluate
      1. 8(5u + 12)
      2. 8 · 5u + 8 · 12
    3. If n = \(\dfrac{2}{3}\), evaluate
      1. \(3 \left(n + \dfrac{5}{6}\right)\)
      2. 3 • n + 3 • \(\dfrac{5}{6}\)
    4. If y = 3 4 , evaluate
      1. 4 ⎛ ⎝ y + 3 8 ⎞ ⎠
      2. 4 • y + 4 • \(\dfrac{3}{8}\)
    5. If y = \(\dfrac{7}{12}\), evaluate
      1. −3(4y + 15)
      2. 3 • 4y + (−3) • 15
    6. If p = \(\dfrac{23}{30}\), evaluate
      1. −6(5p + 11)
      2. −6 • 5p + (−6) • 11
    7. If m = 0.4, evaluate
      1. −10(3m − 0.9)
      2. −10 • 3m − (−10)(0.9)
    8. If n = 0.75, evaluate
      1. −100(5n + 1.5)
      2. −100 • 5n + (−100)(1.5)
    9. If y = −25, evaluate
      1. −(y − 25)
      2. −y + 25
    10. If w = −80, evaluate
      1. −(w − 80)
      2. −w + 80
    11. If p = 0.19, evaluate
      1. −(p + 0.72)
      2. −p − 0.72
    12. If q = 0.55, evaluate
      1. −(q + 0.48)
      2. −q − 0.48

    Everyday Math

    1. Buying by the case Joe can buy his favorite ice tea at a convenience store for $1.99 per bottle. At the grocery store, he can buy a case of 12 bottles for $23.88.
      1. Use the distributive property to find the cost of 12 bottles bought individually at the convenience store. (Hint: notice that $1.99 is $2 − $0.01.)
      2. Is it a bargain to buy the iced tea at the grocery store by the case?
    2. Multi-pack purchase Adele’s shampoo sells for $3.97 per bottle at the drug store. At the warehouse store, the same shampoo is sold as a 3-pack for $10.49.
      1. Show how you can use the distributive property to find the cost of 3 bottles bought individually at the drug store.
      2. How much would Adele save by buying the 3-pack at the warehouse store?

    Writing Exercises

    1. Simplify \(8 \left(x − \dfrac{1}{4}\right)\) using the distributive property and explain each step.
    2. Explain how you can multiply 4($5.97) without paper or a calculator by thinking of $5.97 as 6 − 0.03 and then using the distributive property.

    Self Check

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

    7.4: Distributive Property (13)

    (b) What does this checklist tell you about your mastery of this section? What steps will you take to improve?

    Contributors and Attributions

    7.4: Distributive Property (2024)

    FAQs

    What is the distributive property of 7? ›

    Solution: When we solve the expression 7(20 + 3) using the distributive property, we first multiply every addend by 7. This is known as distributing the number 7 amongst the two addends and then we can add the products. This means that the multiplication of 7(20) and 7(3) will be performed before the addition.

    What is the distributive property of 3x6? ›

    3 x 6 = 3(6) = 3 • 6. 3 • (2 + 4) = 3 • 6 = 18. The distributive property of multiplication over addition can be used when you multiply a number by a sum.

    What are the rules of distributive property? ›

    distributive law, also called distributive property, in mathematics, the law relating the operations of multiplication and addition, stated symbolically as a(b + c) = ab + ac; that is, the monomial factor a is distributed, or separately applied, to each term of the binomial factor b + c, resulting in the product ab + ...

    What is the distributive property for kids? ›

    The property states that whether the numbers in parentheses are added before or after multiplication, the results are the same. The distributive property is especially useful when the polynomial cannot be added together prior to multiplication: 2(3 + 4x)= 6 + 8x.

    What is the distributive property of 7x8? ›

    Distributive Property: This is one of the most important ideas in multiplication. For example, 7x8 can be solved by decomposing 7 into 5 and 2 and making two partial products - 5x8 and 2x8.

    What is the distributive property of 7x6? ›

    First, we can break down 6 into two smaller numbers, like 3 and 3. So, 7x6 = 7x(3+3). Now, we can apply the distributive property: 7x(3+3) = (7x3) + (7x3). So, the number sentence is: 7x6 = (7x3) + (7x3).

    What property is 3 7 7? ›

    Answer and Explanation:

    So here commutative property of natural numbers under addition is shown.

    What is the distributive property of multiplication over addition Class 7? ›

    The distributive property of multiplication over addition is applied when you multiply a value by a sum. For example, you want to multiply 5 by the sum of 10 + 3. As we have like terms, we usually first add the numbers and then multiply by 5. But, according to the property, you can first multiply every addend by 5.

    What is the distributive property of multiplication of integers Class 7? ›

    Distributive Property of Multiplication Over Addition:

    In case of any three integers x, y and z, x × (y + z) = (x × y) + (x × z). = – 24. Distributive Property of Multiplication Over Subtraction: In case of any three integers, x, y and z, x × (y - z) = (x × y) – (x × z).

    What is the property of 7 1 )= 7? ›

    Identity property of multiplication: The product of 1 and any number is that number. For example, 7 × 1 = 7 7 \times 1 = 7 7×1=77, times, 1, equals, 7.

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