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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.

Nígbà yìí Kò ní kọ́ǹpútà alágbèéká kan tí o lè fi àwọn à

Àwọn Ìṣàmúlò-ètò (40)

Wá gbogbo àwọn ní pàtó nínú àwọn àkọ́lé ní ìbẹrẹ. Fi àwọn

Multiply: $3(0.25).$

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

7x − 42
Simplify: $8(x-5).$

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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: $r(s-2).$

Fi àwọn àgbèwọlé hàn

rs − 2r
Simplify: $y(z-8).$

Fi àwọn àgbèwọlé hàn

yz − 8y
Simplify: $(x+8)p.$

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

−18m − 15
Simplify: $-6(8n+11).$

Fi àwọn àgbèwọlé hàn

−48n − 66
Simplify: $-11(4-3a).$

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

−10 + 15a
Simplify: $-7(8-15y).$

Fi àwọn àgbèwọlé hàn

−56 + 105y
Simplify: $-(y+5).$

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

−z + 11
Simplify: $-(x-4).$

Fi àwọn àgbèwọlé hàn

−x + 4
Simplify: $8-2(x+3).$

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

−3x + 3
Simplify: $7x-5(x+4).$

Fi àwọn àgbèwọlé hàn

2x − 20
Simplify: $4(x-8)-(x+3).$

Fi àwọn àgbèwọlé hàn

Distribute.
Combine like terms.

Fi iṣẹ́ rẹ̀ pamọ́

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Bii o ṣe le: Distributive Property

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

Àwọn Àtòjọ-ẹ̀yàn

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.

Àwọn ààyè ojú-ìwé yìí tí a fi pamọ́ láti OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Tí a fi pamọ́ síì níbẹ̀; àwọn àwọn àṣiṣe ní wa. Nítorí pé a tí fi àkóónú rẹ̀ pamọ́ fún ìlòjónú tí kò jẹ́ iṣẹ́ ọ̀fẹ́, ìwé-ìwé yìí jẹ́ ọ̀fẹ̀, kò ní ìfilọ́wọ́, àti àwọn tí a ti pamọ́ fún ìlòjónú kan.

Diẹ̀ nínú 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.