Properties of Operations and the Distributive Property | Grade 8 Math | LevelKite

The expressions $3 + x$ and $x + 3$ look different. Try some values. When $x = 5$ , both equal $8$ . When $x = -2$ , both equal $1$ . In fact, they match for every real number you could choose.

Two expressions are equivalent when they have the same value for every value the variable is allowed to take. Those allowed values are the variable's domain. In this lesson the domain is all real numbers, because every expression here is defined for every real number.

You cannot test every real number. Instead, you use properties: facts about addition and multiplication that are true for all real numbers. When you rewrite an expression by using a property correctly, the new expression is equivalent to the old one.

Numbers that you add are called addends. Numbers that you multiply are called factors. The first properties come from four simple ideas.

Order. You can swap two addends, or swap two factors, and the result stays the same. This is the commutative property.
Grouping. When you add three numbers, or multiply three numbers, you can combine the first two first or the last two first. The result is the same. This is the associative property.
No change. Adding $0$ leaves a number unchanged, and so does multiplying by $1$ . These are the identity properties.
Zero. Multiplying any number by $0$ gives $0$ . This is the zero property of multiplication.

The table states each property in symbols. The letters $a$ , $b$ and $c$ stand for any real numbers.

Each property is true for all real numbers a, b and c.

Property	What it means	In symbols	With numbers
Commutative property of addition	You can add in either order.	$a + b = b + a$	$7 + 5 = 5 + 7$ ; both equal $12$
Commutative property of multiplication	You can multiply in either order.	$a \cdot b = b \cdot a$	$7 \cdot 5 = 5 \cdot 7$ ; both equal $35$
Associative property of addition	You can group addends either way.	$(a + b) + c = a + (b + c)$	$(2 + 3) + 4 = 2 + (3 + 4)$ ; both equal $9$
Associative property of multiplication	You can group factors either way.	$(a \cdot b) \cdot c = a \cdot (b \cdot c)$	$(2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4)$ ; both equal $24$
Identity property of addition	Adding $0$ leaves a number unchanged.	$a + 0 = a$	$9 + 0 = 9$
Identity property of multiplication	Multiplying by $1$ leaves a number unchanged.	$a \cdot 1 = a$	$9 \cdot 1 = 9$
Zero property of multiplication	Multiplying by $0$ gives $0$ .	$a \cdot 0 = 0$	$9 \cdot 0 = 0$

The table leaves out subtraction and division on purpose. With those operations, changing the order or the grouping can change the result:

$8 - 3 = 5$ , but $3 - 8 = -5$ .
$(8 - 3) - 2 = 3$ , but $8 - (3 - 2) = 7$ .
$8 \div 2 = 4$ , but $2 \div 8 = \frac{1}{4}$ .

So the commutative and associative properties do not apply to subtraction or division.

You can still work with a subtraction. Subtracting a number gives the same result as adding its opposite, and the opposite of $5$ is $-5$ . So $x - 5 = x + (-5)$ . Now it is an addition, and you may change the order: $x + (-5) = -5 + x$ . These three expressions are equivalent for every real number $x$ .

Worked example

Name the property behind each step

Show that $5 \cdot (x \cdot 4)$ is equivalent to $20x$ , where $x$ is any real number. Name the property that justifies each step.

Swap the two factors inside the parentheses. The commutative property of multiplication says that changing the order of two factors does not change the product, so the value is the same for every real number x.
$5 \cdot (x \cdot 4) = 5 \cdot (4 \cdot x)$
Move the parentheses so that 5 and 4 are grouped together. The associative property of multiplication says that changing the grouping of factors does not change the product.
$5 \cdot (4 \cdot x) = (5 \cdot 4) \cdot x$
Multiply 5 by 4 to get 20. Writing 20 times x as 20x is just a shorter way to write the same product.
$(5 \cdot 4) \cdot x = 20 \cdot x = 20x$
Check one value, x = 3. Both expressions give 60. One matching value supports the work but does not prove it. The two properties do that, because they are true for every real number.
$5 \cdot (3 \cdot 4) = 5 \cdot 12 = 60 \quad\text{and}\quad 20 \cdot 3 = 60$

Answer

The expression $5 \cdot (x \cdot 4)$ is equivalent to $20x$ for every real number $x$ , by the commutative and associative properties of multiplication.

The properties in the table each use one operation. The distributive property links two operations: multiplication and addition.

Try it with numbers first. To find $6(10 + 4)$ , you can add inside the parentheses first: $6 \cdot 14 = 84$ . Or you can multiply each addend by $6$ and then add: $6 \cdot 10 + 6 \cdot 4 = 60 + 24 = 84$ . Both ways give $84$ .

To see why, read $2(x + 3)$ as two copies of $x + 3$ added together. Addition allows any order and any grouping, so you can put the two $x$ 's together and the two $3$ 's together. Two $x$ 's make $2x$ , and two $3$ 's make $2 \cdot 3 = 6$ .

2(x + 3) = (x + 3) + (x + 3) = (x + x) + (3 + 3) = 2x + 6

Key concept

The distributive property

The parts of a sum are its terms: the terms of $3x + 7$ are $3x$ and $7$ . To multiply a sum by a number, multiply each term by that number, then add the products. Doing this is called distributing the factor.

The picture of copies works only for a whole-number factor, but the property itself is true for all real numbers $a$ , $b$ and $c$ , including negative numbers and fractions. Like every property in this lesson, it gives an equivalent expression: both sides have the same value for every value of the variables.

When the parentheses hold a subtraction, first write it as adding the opposite. That gives $a(b - c) = ab - ac$ .

a(b + c) = ab + ac

Worked example

Distribute a positive factor

Use the distributive property to multiply $4(3x + 7)$ , where $x$ is any real number.

The sum inside the parentheses has two terms, 3x and 7. Multiply each term by the outside factor, 4. The distributive property says the result has the same value as the original expression for every real number x.
$4(3x + 7) = 4 \cdot 3x + 4 \cdot 7$
Find the first product. 3x means 3 times x, so the associative property of multiplication lets you group the 4 with the 3 and multiply those two numbers first.
$4 \cdot 3x = (4 \cdot 3) \cdot x = 12x$
Find the second product. This one is plain arithmetic.
$4 \cdot 7 = 28$
Write the two products as a sum.
$4(3x + 7) = 12x + 28$
Check one value, x = 2. Both expressions give 52. The match supports the work, and the distributive property is what guarantees equal values for every real number x.
$4(3 \cdot 2 + 7) = 4 \cdot 13 = 52 \quad\text{and}\quad 12 \cdot 2 + 28 = 24 + 28 = 52$

Answer

The product $4(3x + 7)$ equals $12x + 28$ for every real number $x$ .

Worked example

Distribute a negative factor

Use the distributive property to multiply $-3(2x - 5)$ , where $x$ is any real number.

Rewrite the subtraction as adding the opposite. Now the inside is a sum and you can see its two terms: 2x and negative 5. The negative sign belongs to the 5.
$-3(2x - 5) = -3(2x + (-5))$
Multiply each term by the outside factor, negative 3, sign included. The distributive property is true for every real number, so it works for a negative factor too.
$-3(2x + (-5)) = (-3)(2x) + (-3)(-5)$
Find the first product by grouping the two numbers. Negative 3 times 2 is negative 6, because a negative number times a positive number is negative.
$(-3)(2x) = (-3 \cdot 2) \cdot x = -6x$
Find the second product. Negative 3 times negative 5 is positive 15, because a negative number times a negative number is positive.
$(-3)(-5) = 15$
Write the two products as a sum.
$-3(2x - 5) = -6x + 15$
Check one value, x = 1. Both expressions give 9. One value only supports the work; the distributive property is what makes the two expressions match for every real number x.
$-3(2 \cdot 1 - 5) = -3(-3) = 9 \quad\text{and}\quad -6 \cdot 1 + 15 = -6 + 15 = 9$

Answer

The product $-3(2x - 5)$ equals $-6x + 15$ for every real number $x$ .

Here is the idea to keep. A property of addition or multiplication is true for all real numbers, so rewriting with a property gives an equivalent expression: one with the same value for every value in the variable's domain.

The commutative property lets you change the order of addends or of factors.
The associative property lets you change the grouping of addends or of factors.
The distributive property multiplies every term inside the parentheses by the factor outside, and each term keeps its own sign.

A test value can catch a mistake, but one matching value does not prove equivalence. The properties do, because they hold for every real number.

Next, you will add and subtract like terms to simplify linear expressions.