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Distributive Property

Simplify expressions using the distributive property

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need $\text{\$9.25};$ 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.

They need $3$ times $\text{\$9},$ so $\text{\$27},$ and $3$ times $1$ quarter, so $75$ cents. In total, they need $\text{\$27.75}.$

If you think about doing the math in this way, you are using the Distributive Property.

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:

\[\begin{array}{lll} & 3(9.25) & \\ 3(9 & + & 0.25) \\ 3(9) & + & 3(0.25) \\ 27 & + & 0.75 \\ & 27.75 & \end{array}\]

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,

Example

Try it.

Simplify: $3(x+4).$

Solution

	$3(x+4)$
Distribute.	$3\cdot x+3\cdot 4$
Multiply.	$3x+12$

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

Example

Try it.

Simplify: $6(5y+1).$

Solution


Distribute.
Multiply.

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

Example

Try it.

Simplify: $2(x-3).$

Solution


Distribute.
Multiply.

Example

Try it.

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

Solution


Distribute.
Simplify.

Example

Try it.

Simplify: $8(\frac{3}{8}x+\frac{1}{4}).$

Solution


Distribute.
Multiply.

Example

Try it.

Simplify: $100(0.3+0.25q).$

Solution


Distribute.
Multiply.

Condensed: the full section is in OpenStax Prealgebra 2e.

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 ⓐ , we will evaluate the form with parentheses, and in part ⓑ we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example

Try it.

When $y=10$ evaluate: ⓐ $\ 6(5y+1)$ ⓑ $\ 6\cdot 5y+6\cdot 1.$

Solution

ⓐ
	$6(5y+1)$

Simplify in the parentheses.	$6(51)$
Multiply.	$306$

ⓑ


Simplify.
Add.

Notice, the answers are the same. When $y=10,$

\[6(5y+1)=6\cdot 5y+6\cdot 1.\]

Try it yourself for a different value of $y.$

Example

Try it.

When $y=3,$ evaluate ⓐ $\ -2(4y+1)$ⓑ $\ -2\cdot 4y+(-2)\cdot 1.$

Solution

ⓐ
	$\ -2(4y+1)$

Simplify in the parentheses.	$-2(13)$
Multiply.	$-26$

ⓑ
	$\ -2\cdot 4y+(-2)\cdot 1$

Multiply.	$-24-2$
Subtract.	$-26$
The answers are the same. When $y=3,$	$-2(4y+1)=-8y-2$

Example

Try it.

When $y=35$ evaluate ⓐ $\ \text{-}(y+5)$ and ⓑ $\ \text{-}\text{y}-5$ to show that $-(y+5)=\text{-}\text{y}-5.$

Solution

ⓐ
	$\ \text{-}(y+5)$

Add in the parentheses.	$-(40)$
Simplify.	$-40$

ⓑ
	$\ -y-5$

Simplify.	$-40$
The answers are the same when $y=35,$ demonstrating that	$-(y+5)=\text{-}y-5$

Distributive Property

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

Try it.

$\frac{1}{2}(n+8)$

Try it.

$\frac{1}{3}(u+9)$

Solution

$\frac{1}{3}u+3$

Try it.

$\frac{1}{4}(3q+12)$

Try it.

$\frac{1}{5}(4m+20)$

Solution

$\frac{4}{5}m+4$

Try it.

$9(\frac{5}{9}y-\frac{1}{3})$

Try it.

$10(\frac{3}{10}x-\frac{2}{5})$

Solution

3x − 4

Try it.

$12(\frac{1}{4}+\frac{2}{3}r)$

Try it.

$12(\frac{1}{6}+\frac{3}{4}s)$

Solution

2 + 9s

Try it.

$-2(y+13)$

Try it.

$-3(a+11)$

Solution

−3a − 33

Try it.

$-7(4p+1)$

Try it.

$-9(9a+4)$

Solution

−81a − 36

Try it.

$-9(3a-7)$

Try it.

$-6(7x-8)$

Solution

−42x + 48

Try it.

$5+9(n-6)$

Try it.

$12+8(u-1)$

Solution

8u + 4

Try it.

$16-3(y+8)$

Try it.

$18-4(x+2)$

Solution

−4x + 10

Try it.

$4-11(3c-2)$

Try it.

$9-6(7n-5)$

Solution

−42n + 39

Try it.

$22-(a+3)$

Try it.

$-12-(u+10)$

Try it.

$-4-(c-10)$

Solution

−c + 6

Try it.

$(5m-3)-(m+7)$

Try it.

$(4y-1)-(y-2)$

Solution

3y + 1

Try it.

$5(2n+9)+12(n-3)$

Try it.

$9(5u+8)+2(u-6)$

Solution

47u + 60

Try it.

$9(8x-3)-(-2)$

Try it.

$4(6x-1)-(-8)$

Solution

24x + 4

Try it.

$14(c-1)-8(c-6)$

Try it.

$11(n-7)-5(n-1)$

Solution

6n − 72

Try it.

$6(7y+8)-(30y-15)$

Try it.

$7(3n+9)-(4n-13)$

Solution

17n + 76

Evaluate Expressions Using the Distributive Property

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

Try it.

If $v=-2,$ evaluate

ⓐ $\ 6(4v+7)$
ⓑ $\ 6\cdot 4v+6\cdot 7$

Try it.

If $u=-1,$ evaluate

ⓐ $\ 8(5u+12)$
ⓑ $\ 8\cdot 5u+8\cdot 12$

Solution

ⓐ 56
ⓑ 56

Try it.

If $n=\frac{2}{3},$ evaluate

ⓐ $\ 3(n+\frac{5}{6})$
ⓑ $\ 3\cdot n+3\cdot \frac{5}{6}$

Try it.

If $y=\frac{3}{4},$ evaluate

ⓐ $\ 4(y+\frac{3}{8})$
ⓑ $\ 4\cdot y+4\cdot \frac{3}{8}$

Solution

ⓐ $\ \frac{9}{2}$
ⓑ $\ \frac{9}{2}$

Try it.

If $y=\frac{7}{12},$ evaluate

ⓐ $\ -3(4y+15)$
ⓑ $\ -3\cdot 4y+(-3)\cdot 15$

Try it.

If $p=\frac{23}{30},$ evaluate

ⓐ $\ -6(5p+11)$
ⓑ $\ -6\cdot 5p+(-6)\cdot 11$

Solution

ⓐ −89
ⓑ −89

Try it.

If $m=0.4,$ evaluate

ⓐ $\ -10(3m-0.9)$
ⓑ $\ -10\cdot 3m-(-10)(0.9)$

Try it.

If $n=0.75,$ evaluate

ⓐ $\ -100(5n+1.5)$
ⓑ $\ -100\cdot 5n+(-100)(1.5)$

Solution

ⓐ −525
ⓑ −525

Try it.

If $y=-25,$ evaluate

ⓐ $\ -(y-25)$
ⓑ $\ -y+25$

Try it.

If $w=-80,$ evaluate

ⓐ $\ -(w-80)$
ⓑ $\ -w+80$

Solution

ⓐ 160
ⓑ 160

Try it.

If $p=0.19,$ evaluate

ⓐ $\ -(p+0.72)$
ⓑ $\ -p-0.72$

Try it.

If $q=0.55,$ evaluate

ⓐ $\ -(q+0.48)$
ⓑ $\ -q-0.48$

Solution

ⓐ −1.03
ⓑ −1.03

Try it.

Simplify $8(x-\frac{1}{4})$ using the distributive property and explain each step.

Try it.

Explain how you can multiply $4(\text{\$5.97})$ without paper or a calculator by thinking of $\text{\$5.97}$ as $6-0.03$ and then using the distributive property.

Solution

Answers will vary.

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

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

Condensed: the full section is in OpenStax Prealgebra 2e.

이제 너 계산기는 이것을 해결하지 않지만, 그 조각은 계산 가능합니다. 아래의 하나를 시도하거나 자신의 것을 입력하십시오.

연습 (40)

먼저 종이에 각각을 시도해 보세요. 확인하기 위해 답을 드러내세요. 확인된 답은 각 단계의 솔버에서 열 수 있습니다.

Multiply: $3(0.25).$

답을 드러내세요

$0.75$
Simplify: $10-(-2)(3).$

답을 드러내세요

$16$
Combine like terms: $9y+17+3y-2.$

답을 드러내세요

$12y+15$
Simplify: $3(x+4).$

답을 드러내세요

$3(x+4)$
Distribute. $3\cdot x+3\cdot 4$
Multiply. $3x+12$
Simplify: $4(x+2).$

답을 드러내세요

4x + 8
Simplify: $6(x+7).$

답을 드러내세요

6x + 42
Simplify: $6(5y+1).$

답을 드러내세요

Distribute.
Multiply.
Simplify: $9(3y+8).$

답을 드러내세요

27y + 72
Simplify: $5(5w+9).$

답을 드러내세요

25w + 45
Simplify: $2(x-3).$

답을 드러내세요

Distribute.
Multiply.
Simplify: $7(x-6).$

답을 드러내세요

7x − 42
Simplify: $8(x-5).$

답을 드러내세요

8x − 40
Simplify: $\frac{3}{4}(n+12).$

답을 드러내세요

Distribute.
Simplify.
Simplify: $\frac{2}{5}(p+10).$

답을 드러내세요

$\frac{2}{5}p+4$
Simplify: $\frac{3}{7}(u+21).$

답을 드러내세요

$\frac{3}{7}u+9$
Simplify: $8(\frac{3}{8}x+\frac{1}{4}).$

답을 드러내세요

Distribute.
Multiply.
Simplify: $6(\frac{5}{6}y+\frac{1}{2}).$

답을 드러내세요

5y + 3
Simplify: $12(\frac{1}{3}n+\frac{3}{4}).$

답을 드러내세요

4n + 9
Simplify: $100(0.3+0.25q).$

답을 드러내세요

Distribute.
Multiply.
Simplify: $100(0.7+0.15p).$

답을 드러내세요

70 + 15p
Simplify: $100(0.04+0.35d).$

답을 드러내세요

4 + 35d
Simplify: $m(n-4).$

답을 드러내세요

Distribute.
Multiply.
Notice that we wrote $m\cdot 4\ \text{as}\ 4m.$ 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.
Simplify: $r(s-2).$

답을 드러내세요

rs − 2r
Simplify: $y(z-8).$

답을 드러내세요

yz − 8y
Simplify: $(x+8)p.$

답을 드러내세요

Distribute.
Simplify: $(x+2)p.$

답을 드러내세요

xp + 2p
Simplify: $(y+4)q.$

답을 드러내세요

yq + 4q
Simplify: $-2(4y+1).$

답을 드러내세요

Distribute.
Simplify.
Simplify: $-3(6m+5).$

답을 드러내세요

−18m − 15
Simplify: $-6(8n+11).$

답을 드러내세요

−48n − 66
Simplify: $-11(4-3a).$

답을 드러내세요

Distribute.
Multiply.
Simplify.
You could also write the result as $33a-44.$ Do you know why?
Simplify: $-5(2-3a).$

답을 드러내세요

−10 + 15a
Simplify: $-7(8-15y).$

답을 드러내세요

−56 + 105y
Simplify: $-(y+5).$

답을 드러내세요

Multiplying by −1 results in the opposite.
Distribute.
Simplify.
Simplify.
Simplify: $-(z-11).$

답을 드러내세요

−z + 11
Simplify: $-(x-4).$

답을 드러내세요

−x + 4
Simplify: $8-2(x+3).$

답을 드러내세요

Distribute.
Multiply.
Combine like terms.
Simplify: $9-3(x+2).$

답을 드러내세요

−3x + 3
Simplify: $7x-5(x+4).$

답을 드러내세요

2x − 20
Simplify: $4(x-8)-(x+3).$

답을 드러내세요

Distribute.
Combine like terms.

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어떻게: Distributive Property

Simplify expressions using the distributive property
Evaluate expressions using the distributive property
If

사람들이 묻는 질문

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

이 페이지의 일부는 다음에서 변경되었습니다. OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). 여기서 압축하고 다시 설명; 오류는 우리의. 소스가 비상업적 사용을 위해 라이선스되어 있기 때문에, 이 레슨은 무료이며, 아무것도 판매하지 않으며, 동일한 라이선스로 공유됩니다.

에 더 Arithmetic

Order of operations Fractions Percentages Powers and roots

	\(3(x+4)\)
Distribute.	\(3\cdot x+3\cdot 4\)
Multiply.	\(3x+12\)

ⓑ
	\(\ -2\cdot 4y+(-2)\cdot 1\)

Multiply.	\(-24-2\)
Subtract.	\(-26\)
The answers are the same. When \(y=3,\)	\(-2(4y+1)=-8y-2\)

ⓑ
	\(\ -y-5\)

Simplify.	\(-40\)
The answers are the same when \(y=35,\) demonstrating that	\(-(y+5)=\text{-}y-5\)

	\(3(x+4)\)
Distribute.	\(3\cdot x+3\cdot 4\)
Multiply.	\(3x+12\)

ⓐ
	\(6(5y+1)\)

Simplify in the parentheses.	\(6(51)\)
Multiply.	\(306\)

ⓐ
	\(\ -2(4y+1)\)

Simplify in the parentheses.	\(-2(13)\)
Multiply.	\(-26\)

ⓐ
	\(\ \text{-}(y+5)\)

Add in the parentheses.	\(-(40)\)
Simplify.	\(-40\)


Multiplying by −1 results in the opposite.
Distribute.
Simplify.
Simplify.