Pre-Algebra · Grades 7

Combining Like Terms

Quick answer

Like terms have exactly the same variable part, so 3x and 5x are alike while 3x and 3y are not. Collecting them means adding the coefficients and leaving the variable alone: 3x + 5x is 8x, not 8x². The variable is a label saying what is being counted, and counting eight of something does not change what it is.

What you'll learn

  • Identify which terms in an expression are alike
  • Combine like terms by adding their coefficients
  • Move terms with their signs attached

Like terms have the same variable part

3x+5x=8x3x + 5x = 8x

3x3x and 5x5x are like terms: their variable parts are identical. Only the coefficients differ, and those are what get added.

PairAlike?Why
3x3x, 5x5xyessame variable
3x3x, 3y3ynodifferent letters
3x3x, 3x23x^2nodifferent powers
4xy4xy, 4yx4yxyesxyxy and yxyx are the same product
77, −2-2yesboth constants

The last two rows are worth noting. xyxy and yxyx match because multiplication is commutative, and constants are alike with each other because they share the same (empty) variable part.

Why the variable does not change

3x+5x=8xnot8x23x + 5x = 8x \qquad \text{not} \qquad 8x^2

Think of the variable as a label saying what is being counted.

Three apples plus five apples is eight apples. It is not eight apple-squared, and it is not eight of something else. Counting more of a thing does not change what the thing is.

3 apples+5 apples=8 apples3\text{ apples} + 5\text{ apples} = 8\text{ apples} 3x+5x=8x3x + 5x = 8x

The same reasoning explains why unlike terms cannot be combined. Three apples plus five oranges is not eight of anything — you can only write 3a+5o3a + 5o and leave it.

3x+5ystays as it is3x + 5y \quad \text{stays as it is}

Signs travel with their terms

Every term carries the sign written in front of it.

7x−4+2x+97x - 4 + 2x + 9

The four terms are 7x7x, −4-4, 2x2x and +9+9. Reordering is safe as long as each sign goes along:

7x+2x−4+9=9x+57x + 2x - 4 + 9 = 9x + 5

That reordering is legitimate because subtraction has effectively been rewritten as adding a negative, and addition is commutative. Moving a term while leaving its sign behind is the most common way this goes wrong.

Working through a longer expression

5a−3b+2a+7b−45a - 3b + 2a + 7b - 4

Group the like terms, keeping every sign attached:

(5a+2a)+(−3b+7b)−4(5a + 2a) + (-3b + 7b) - 4

Then add the coefficients within each group:

7a+4b−47a + 4b - 4

Three unlike groups remain, so the expression is now as simple as it gets.

Why simplifying first is worth the step

An expression in its collected form is shorter, and that is the least of it.

It makes the structure visible. 9x+59x + 5 says plainly that the expression is a rate of 99 plus a fixed 55. The original 7x−4+2x+97x - 4 + 2x + 9 says the same thing and hides it.

It makes solving possible. An equation cannot be solved while the variable appears in several places. Collecting like terms is the step that turns 7x−4+2x=237x - 4 + 2x = 23 into 9x−4=239x - 4 = 23, which can then be solved by ordinary means.

It reduces the work. Evaluating 9x+59x + 5 at six different values takes six short calculations. Evaluating the uncollected version takes six long ones, with six chances to slip.

Worked examples

Common mistakes

Practice problems

  1. Simplify 2x+6x2x + 6x.

    Answer

    8x8x

    Full solution

    (2+6)x=8x(2 + 6)x = 8x.

  2. Simplify 9a−4a9a - 4a.

    Answer

    5a5a

    Full solution

    (9−4)a=5a(9 - 4)a = 5a.

  3. Simplify 3m+5n+2m3m + 5n + 2m.

    Answer

    5m+5n5m + 5n

    Full solution

    Only the mm terms are alike.

  4. Simplify 4x+3+2x+84x + 3 + 2x + 8.

    Answer

    6x+116x + 11

    Full solution

    (4x+2x)+(3+8)=6x+11(4x + 2x) + (3 + 8) = 6x + 11.

  5. Simplify 7p−2−3p+97p - 2 - 3p + 9.

    Answer

    4p+74p + 7

    Full solution

    (7p−3p)+(−2+9)=4p+7(7p - 3p) + (-2 + 9) = 4p + 7.

  6. Can 5x2+2x5x^2 + 2x be simplified?

    Answer

    No

    Full solution

    x2x^2 and xx are different variable parts, so the terms are not alike.

  7. Simplify 3ab+5ba3ab + 5ba.

    Hint

    Does the order of the letters matter?

    Answer

    8ab8ab

    Full solution

    Multiplication is commutative, so baba is the same as abab. The terms are alike and their coefficients add.

  8. Simplify 6y−10y6y - 10y.

    Answer

    −4y-4y

    Full solution

    (6−10)y=−4y(6 - 10)y = -4y.

  9. Simplify x+x+xx + x + x.

    Answer

    3x3x

    Full solution

    Each xx has coefficient 11, so the coefficients add to 33.

    Note that x⋅x⋅xx \cdot x \cdot x would be x3x^3 — adding and multiplying give very different results here.

  10. Asked to simplify 4x+3x4x + 3x, Omar writes 7x27x^2. Find his error.

    Hint

    What operation is being done to the coefficients?

    Answer

    He changed the variable. The answer is 7x7x.

    Full solution

    Adding like terms adds the coefficients and leaves the variable part untouched: (4+3)x=7x(4 + 3)x = 7x.

    Omar added the coefficients and then multiplied the variables, borrowing a step from 4x×3x4x \times 3x, which really would give 12x212x^2.

    Testing a value catches it. At x=2x = 2: 4(2)+3(2)=144(2) + 3(2) = 14, and 7(2)=147(2) = 14. His 7x27x^2 gives 7(4)=287(4) = 28, which is twice too big.

Frequently asked questions

What are like terms?

Terms with exactly the same variable part. 3x and 5x are alike; 3x and 3y are not, and neither are 3x and 3x².

How do I combine like terms?

Add the coefficients and keep the variable part unchanged. 3x + 5x = 8x.

Why does 3x + 5x not equal 8x²?

Because you are counting, not multiplying. Three x's plus five x's is eight x's, in the same way three apples plus five apples is eight apples.

Are 4xy and 4yx like terms?

Yes. Multiplication is commutative, so xy and yx are the same variable part written two ways.

What happens to the sign when I move a term?

It travels with the term. In 7x - 4 + 2x, the -4 keeps its minus wherever it goes.

What to learn next

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.7.EE.A.1Expressions and EquationsApply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
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