Math Core

Suppose a store receipt lists "3 notebooks, 2 pens, 4 notebooks, 5 pens." You'd naturally shorten that to "7 notebooks, 7 pens." Combining like terms does the same thing for expressions: it groups matching pieces so the expression is shorter and easier to use.

Terms, coefficients and constants

An expression like $5x - 3 + 2x$ is built from terms, the pieces joined by $+$ or $-$ signs. Here the terms are $5x$ , $-3$ and $2x$ .

The number multiplied by a variable is its coefficient. In $5x$ the coefficient is $5$ . In $x$ the coefficient is $1$ , and in $-x$ it is $-1$ .
A term with no variable, like $-3$ , is a constant.

Each term owns the sign in front of it. In $8 - 6y$ , the terms are $8$ and $-6y$ , so the coefficient of $y$ is $-6$ , not $6$ .

Definition

Like terms

Like terms have exactly the same variable part. Only their coefficients can be different. All constants are like terms with each other.

like terms	not like terms	why not
$4a$ and $-9a$	$4a$ and $4b$	different variables
$\tfrac{1}{2}m$ and $3m$	$3m$ and $3$	one has no variable
$7$ and $-2.5$	$2xy$ and $2x$	$2x$ has no $y$

Why you add the coefficients

Think of $x$ as a mystery box. Then $5x$ means $5$ boxes and $2x$ means $2$ more boxes. Together that's $7$ boxes: $5x + 2x = 7x$ . The distributive property says the same thing:

5x + 2x = (5 + 2)x = 7x.

But $5x + 2y$ is $5$ of one thing and $2$ of something different. You can't shorten it, just like $5$ notebooks and $2$ pens don't make $7$ of anything.

Combining like terms

To combine like terms, add their coefficients and keep the variable part the same. Terms that aren't alike stay separate.

Step by step

Find each group of like terms. Bring the sign in front of each term along with it.
Rearrange so like terms sit next to each other. (Addition can be done in any order, as long as each term keeps its sign.)
Add the coefficients in each group.
Write the variable terms first and the constant last.

Worked example: Two groups

Simplify $6x + 5 - 2x - 8$ .

\begin{aligned} 6x + 5 - 2x - 8 &= 6x - 2x + 5 - 8 && \text{group like terms} \\ &= (6 - 2)x + (5 - 8) && \text{add coefficients} \\ &= 4x - 3 \end{aligned}

Worked example: Negative coefficients and a hidden 1

Simplify $-3n + 7 - n + 4n - 10$ .

The $n$ -terms are $-3n$ , $-n$ (coefficient $-1$ ) and $4n$ . The constants are $7$ and $-10$ .

(-3 - 1 + 4)n + (7 - 10) = 0n - 3 = -3.

The $n$ -terms cancel completely, so the whole expression equals $-3$ no matter what $n$ is.

Fractions and decimals

Rational coefficients work exactly the same way. You just use your fraction and decimal skills to add them.

Worked example: Rational coefficients

Simplify $\dfrac{1}{2}y + 3 + \dfrac{1}{4}y - 1.5$ .

Add the $y$ -coefficients with a common denominator: $\dfrac{1}{2} + \dfrac{1}{4} = \dfrac{2}{4} + \dfrac{1}{4} = \dfrac{3}{4}$ .

Add the constants: $3 - 1.5 = 1.5$ .

\frac{1}{2}y + 3 + \frac{1}{4}y - 1.5 = \frac{3}{4}y + 1.5

Common mistake

Watch out for these mistakes:

Combining unlike terms. $4x + 3$ is already simplified. It is not $7x$ .
Dropping a sign. In $10 - 3x + x$ , the $x$ -terms are $-3x$ and $+x$ , which make $-2x$ . The answer is $10 - 2x$ , not $10 - 4x$ .
Writing $x + x = x^2$ . Adding two $x$ 's gives $2x$ .

Adding and subtracting expressions

To add two expressions, drop the parentheses and combine. To subtract an expression, subtract every term in it. That means each sign inside the parentheses flips.

Worked example: Subtracting an expression

Simplify $(5a + 4) - (2a - 6)$ .

Subtracting $2a$ gives $-2a$ . Subtracting $-6$ is the same as adding $6$ .

\begin{aligned} (5a + 4) - (2a - 6) &= 5a + 4 - 2a + 6 \\ &= (5a - 2a) + (4 + 6) \\ &= 3a + 10 \end{aligned}

Tip

Check your work by plugging in a number. Try $a = 1$ in the last example. Original: $(5 + 4) - (2 - 6) = 9 - (-4) = 13$ . Simplified: $3 + 10 = 13$ . They match.

Practice

Practice 1

Which pair are like terms?

Practice 2

In the expression $9 - 7m + 2$ , what is the coefficient of $m$ ?

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

Practice 3

Simplify $8x + 3 - 5x + 6$ .

Practice 4

Simplify $-4b + 2 - b - 9$ .

Practice 5

After simplifying $\dfrac{1}{3}x + 4 + \dfrac{1}{2}x$ , what is the coefficient of $x$ ?

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

Practice 6

Simplify $2.5t - 1.2 + 0.5t + 3$ .

Practice 7

Simplify $(7y - 2) - (3y - 5)$ .

Practice 8

A triangle has sides of length $x + 4$ , $2x - 1$ and $3x + 4$ centimeters. Its perimeter simplifies to $6x + c$ . What is $c$ ?

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