Math Core

Ana buys 3 packs of 4 red pencils and 3 packs of 2 blue pencils. Ben says she has $3 \times 4 + 3 \times 2$ pencils. Cara says she has $3 \times (4 + 2)$ . They are both right: both expressions equal 18. Expressions that always have the same value are called equivalent, and knowing how to rewrite them is one of the most useful skills in algebra.

What "equivalent" means

Definition

Equivalent expressions

Two expressions are equivalent if they have the same value no matter what number you substitute for the variable.

For example, $x + x + x$ and $3x$ are equivalent. Adding three copies of a number is the same as multiplying it by 3, for every number.

But $2x$ and $x + 2$ are not equivalent. They happen to match when $x = 2$ (both equal 4), but when $x = 5$ you get $2(5) = 10$ and $5 + 2 = 7$ . One value that doesn't match is enough to show two expressions are not equivalent.

Properties you can use

These properties of operations let you rewrite an expression without changing its value.

property	what it says	example
commutative	you can switch the order when adding or multiplying	$a + 5 = 5 + a$ ; $\;4 \cdot n = n \cdot 4$
associative	you can regroup when adding or multiplying	$(y + 2) + 6 = y + (2 + 6)$
distributive	multiplying a sum is the same as multiplying each term	$3(x + 4) = 3x + 12$

The distributive property

a(b + c) = ab + ac \qquad\qquad a(b - c) = ab - ac

The number outside the parentheses multiplies every term inside.

You can picture $3(x + 4)$ as a rectangle that is 3 units tall and $x + 4$ units long. Split it into two smaller rectangles. Their areas are $3x$ and $3 \times 4 = 12$ , so the total area is $3x + 12$ .

An area model for 3(x + 4) = 3x + 12.

Common mistake

Don't forget to multiply the second term. A very common mistake is writing $3(x + 4) = 3x + 4$ . The 3 multiplies the 4, too, so it is $3x + 12$ .

Worked example: Using the distributive property

Use the distributive property to rewrite $5(2y - 3)$ .

Multiply each term inside by 5: $5 \cdot 2y = 10y$ and $5 \cdot 3 = 15$ . So $5(2y - 3) = 10y - 15$ .

Combining like terms

Like terms have exactly the same variable part. In $4x + 7 + 2x + 1$ , the terms $4x$ and $2x$ are like terms, and the constants $7$ and $1$ are like terms. But $4x$ and $7$ are not.

To combine like terms, add their coefficients. Think of $4x + 2x$ as "4 bags of $x$ marbles plus 2 more bags," which makes 6 bags: $6x$ .

Worked example: Combining like terms

Write an equivalent expression with as few terms as possible: $4x + 7 + 2x + 1$ .

Group the like terms (the commutative property lets you move them): $(4x + 2x) + (7 + 1)$ . Then combine: $6x + 8$ .

Worked example: Distribute, then combine

Write $2(3a + 5) + a$ with as few terms as possible.

\begin{aligned} 2(3a + 5) + a &= 6a + 10 + a && \text{distribute} \\ &= 7a + 10 && \text{combine } 6a + a \end{aligned}

Remember that $a$ by itself has a coefficient of 1, so $6a + a = 7a$ .

Factoring: the distributive property in reverse

You can also run the distributive property backward. To factor $12x + 18$ , look for the greatest common factor (GCF) of 12 and 18. It is 6. Then write each term as 6 times something: $12x = 6 \cdot 2x$ and $18 = 6 \cdot 3$ . So

12x + 18 = 6(2x + 3).

Tip

Check any rewrite by substituting a number. For $12x + 18 = 6(2x + 3)$ , try $x = 1$ : the left side is $12 + 18 = 30$ and the right side is $6(5) = 30$ . They match. (Checking one number can catch mistakes, but only the properties can prove two expressions are always equal.)

Practice

Practice 1

Which expression is equivalent to $4(x + 5)$ ?

Practice 2

When you combine like terms in $7y + 3y$ , you get a single term. What is its coefficient?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which expression is equivalent to $6m + 2 + m + 5$ ?

Practice 4

Use the distributive property to rewrite $8(3n - 2)$ in the form $an - b$ . What is the value of $a$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Which pair of expressions is not equivalent?

Practice 6

Write $2(5k + 4) + 5k$ with as few terms as possible. What is the coefficient of $k$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Which expression shows $20x + 30$ factored using the greatest common factor?

Practice 8

The expression $4(2w + 3) + 2(w + 1)$ can be written in the form $aw + b$ . What is $b$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.