Can I combine 5x and 5y since the coefficients are the same?

No. Like terms are decided by the variable part, not the coefficient. 5x and 5y represent different quantities, so 5x + 5y stays as it is. Matching coefficients never make terms combinable.

What happens to the variable when the coefficients add to zero?

The whole term disappears. In 7x - 7x + 3, the variable terms give 0x, and 0 times anything is 0, so the expression simplifies to just 3. Do not carry the x into the constant — writing 3x would be a different expression entirely.

Does the order of my final terms matter?

Not for correctness. 4 + 2x and 2x + 4 are the same expression. Most teachers prefer variable terms first, in alphabetical order, with the constant at the end, because it makes answers easy to compare and easy to check.

How can I be sure my simplified expression is right?

Substitute a number for the variable into both the original and your simplified version. If both give the same value, that is strong evidence they are equivalent. Try a second number, ideally a negative one, since sign errors often hide when you only test positives.

M7MATH-5.3

Combining Like Terms

Learn how to spot like terms in a linear expression and combine them to write a simpler equivalent expression — with sign rules, tables, and worked practice.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Combining Like Terms, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Combining Like Terms free

What this lesson covers

You already know how to write an algebraic expression and how to evaluate one. Now comes the move that makes algebra actually feel shorter: combining like terms. An expression like

7x + 4 - 3x + 9

has four terms, but two of them are describing the same thing, so you can squeeze it down to

4x + 13

without changing its value for any

x

.

The whole skill rests on one question you ask about every pair of terms: do they have exactly the same variable part? If yes, you add or subtract their coefficients. If no, you leave them alone. That sounds simple, and it is — but almost every mistake students make in this lesson comes from mishandling a minus sign or from accidentally combining terms that only look similar. This guide walks through both traps carefully.

Terms, Coefficients, and What Makes Terms "Like"

A term is a single piece of an expression separated by plus or minus signs. In

5x + 8 - 2x

, the terms are

5x

+8

, and

-2x

. Notice that the sign in front travels with the term — that idea will save you later.

Each variable term has two parts. The coefficient is the number multiplied by the variable, and the variable part is the letter (or letters) with its exponent. In

-2x

, the coefficient is

-2

and the variable part is

x

.

Like terms have exactly the same variable part. The coefficients can be anything at all.

Pair	Like terms?	Why
$6x$ and $11x$	Yes	Same variable part, $x$
$6x$ and $11y$	No	Different variables
$6x$ and $11$	No	One has a variable, one does not
$-4a$ and $a$	Yes	Both have variable part $a$ ; the second has coefficient $1$
$9$ and $-2$	Yes	Both are constants

A constant is a term with no variable, like

8

-15

. All constants are like terms with each other, so they can always be combined.

One detail that trips people up: a variable with no visible number in front has a coefficient of

1

, and

-x

means

-1x

. When you combine

5x - x

, you are computing

5 - 1 = 4

, giving

4x

— not

5x

How Combining Works and Why It's Allowed

Combining like terms is not a new rule invented for algebra class. It is the distributive property read backwards, plus the fact that addition can be reordered.

Why does

7x + 3x = 10x

? Because

7x + 3x = (7 + 3)x = 10x

. The

x

is a common factor, so you factor it out, add the numbers, and put the

x

back. That is why only the coefficients change and the variable part stays exactly as it was.

A concrete picture helps. If

x

stands for the weight of one box, then

7

boxes plus

3

boxes is

10

boxes. But

7

boxes plus

3

bags is just

7

boxes and

3

bags — you cannot merge them into one number of anything, which is exactly why

7x + 3y

stays as

7x + 3y

.

The procedure is short:

First, rewrite every subtraction as adding a negative, so each term carries its own sign. Second, group the like terms together. Third, add the coefficients within each group. Fourth, write the result, usually with variable terms first and the constant last.

One critical check: the simplified expression must be equivalent to the original, meaning both give the same value for every substitution. You can test this. In

7x + 4 - 3x + 9

, let

x = 2

: the original is

14 + 4 - 6 + 9 = 21

, and the simplified

4x + 13

gives

8 + 13 = 21

. Matching values for a couple of different inputs is strong evidence you simplified correctly, and it is a habit worth keeping.

Sign Errors — Where Almost Everything Goes Wrong

Most wrong answers in this lesson are not about which terms are alike. They are about minus signs.

The fix is to always attach the sign to the term on its right. In

12 - 5x - 8 + 2x

, the terms are

+12

-5x

-8

, and

+2x

. Now group:

-5x + 2x = -3x

and

12 - 8 = 4

. The answer is

-3x + 4

.

Students who instead read "

5x

and

2x

, so

7x

" get

7x

and lose the whole meaning of the expression. Rewriting the expression with explicit signs before you group takes five seconds and prevents this.

Expression	Common wrong answer	Correct answer	What happened
$8x - 10x$	$2x$	$-2x$	Subtracting a bigger coefficient gives a negative result
$6 - x$	$5x$ or $5$	$6 - x$	Not like terms; nothing to combine
$9y - y$	$9$	$8y$	The variable does not disappear; $9 - 1 = 8$
$4 - 3n + 7n$	$8n$	$4n + 4$	Constant $4$ was combined with variable terms
$-2x + 2x + 5$	$5x$	$5$	The $x$ terms cancel to $0x$ , leaving only $5$

That last row deserves attention. When opposite coefficients cancel, the variable term becomes

0x = 0

and disappears entirely — you do not carry the

x

over to the constant.

Also remember that an expression with no like terms, such as

3a + 7b - 2

, is already simplified. "Simplify" does not promise a shorter answer; it means combine whatever can be combined.

Longer Expressions and Multiple Variables

Real problems often have three or four groups of like terms. Staying organized matters more than being fast.

Simplify

5m + 3n - 2m + 8 - 7n - 1

.

List the terms with signs:

+5m

+3n

-2m

+8

-7n

-1

.

Group them:

(5m - 2m) + (3n - 7n) + (8 - 1)

.

Combine each group:

3m - 4n + 7

.

A useful trick for keeping track is to mark each family of like terms differently — circle the

m

terms, underline the

n

terms, box the constants — then handle one family at a time. Whatever system you use, cross off each term as you use it so nothing gets counted twice or skipped.

Watch out for terms that look alike but are not.

x

and

x^2

are not like terms, because the variable parts differ. In seventh grade most expressions are linear, so you will mostly see first powers, but if

x^2

shows up, treat it as its own family:

4x^2 + 3x - x^2 = 3x^2 + 3x

.

Order does not affect correctness. Both

3m - 4n + 7

and

7 - 4n + 3m

are correct simplifications. Still, writing variable terms in alphabetical order with the constant at the end makes your work easier to check and easier for a classmate or your teacher to read.

In the next lesson you will meet expressions with parentheses, like

3(2x + 5) - 4x

. You will distribute first, then combine like terms — so this skill becomes step two of nearly every simplification you do from here on.

Real Situations Where You Combine Terms

Combining like terms is how you turn a messy description of a situation into a usable formula.

Suppose a school club sells bracelets. Each bracelet costs 3 dollars to make, and the club also pays a 20 dollar table fee at the fair. They then buy extra string for 5 dollars and make an additional design at 2 dollars per bracelet. Total cost with

b

bracelets is

3b + 20 + 2b + 5

. Combining gives

5b + 25

— a formula the club can actually use to budget.

Or consider perimeter. A rectangle has length

x + 4

and width

x

. Perimeter is

(x + 4) + x + (x + 4) + x

, which combines to

4x + 8

. That single expression replaces four separate measurements, and you can evaluate it instantly for any

x

.

Here is the important point about equivalence:

3b + 20 + 2b + 5

and

5b + 25

are two different-looking descriptions of the exact same quantity. The first shows the history of how costs piled up; the second is efficient to compute with. Neither is more "true" than the other. Choosing the simpler equivalent form is what algebra is for — and when you start solving equations, having like terms already combined is what makes the solving possible.

Key terms

Term.: A single part of an expression separated from other parts by addition or subtraction; the sign in front belongs to the term. In $4x - 7$ , the terms are $4x$ and $-7$ .
Like terms.: Terms that have exactly the same variable part, including exponents. $6y$ and $-2y$ are like terms; $6y$ and $6y^2$ are not.
Coefficient.: The number multiplied by the variable in a term. In $-9k$ the coefficient is $-9$ ; in $k$ the coefficient is $1$ .
Constant.: A term with no variable, such as $12$ or $-5$ . Any two constants are like terms.
Equivalent expressions.: Two expressions that produce the same value for every possible value of the variable, such as $7x + 4 - 3x$ and $4x + 4$ .
Simplify.: To rewrite an expression in an equivalent form with as few terms as possible by combining all like terms.
Variable part.: The letter or letters (with exponents) in a term, ignoring the coefficient. The variable part of $-8ab$ is $ab$ .
Distributive property.: The rule $a(b + c) = ab + ac$ , which read in reverse as $ab + ac = a(b + c)$ is the reason like terms can be combined.

Worked example

Simplify the expression

9p - 4 + 2q - 6p + 11 - 5q

. Then check your answer by evaluating both forms at

p = 3

and

q = 2

Step 1 — List each term with its sign attached. Reading left to right:

+9p

-4

+2q

-6p

+11

-5q

. Notice that the minus signs stayed with

4

6p

, and

5q

.

Step 2 — Sort the terms into families. The

p

terms are

+9p

and

-6p

. The

q

terms are

+2q

and

-5q

. The constants are

-4

and

+11

. Every term has been used exactly once — six terms in, six terms sorted.

Step 3 — Combine each family by adding coefficients. For

p

9 - 6 = 3

, giving

3p

. For

q

2 - 5 = -3

, giving

-3q

. For the constants:

-4 + 11 = 7

.

Step 4 — Write the simplified expression, variables first in alphabetical order, constant last:

3p - 3q + 7

.

Step 5 — Check. Substitute

p = 3

and

q = 2

into the original:

9(3) - 4 + 2(2) - 6(3) + 11 - 5(2) = 27 - 4 + 4 - 18 + 11 - 10 = 10

. Now the simplified form:

3(3) - 3(2) + 7 = 9 - 6 + 7 = 10

. Both give

10

, so the expressions are equivalent.

The most likely error here is writing

+3q

instead of

-3q

because

5

is larger than

2

. Keeping the signs attached from Step 1 is what prevents it.

Practice questions

Which expression is equivalent to

8 - 3x + 5x - 12

$2x - 4$
$8x - 4$
$-2x + 20$
$2x + 20$

Answer: $2x - 4$

Sort with signs attached: the

x

terms are

-3x

and

+5x

, and the constants are

+8

and

-12

. For the variables,

-3 + 5 = 2

, giving

2x

. For the constants,

8 - 12 = -4

. So the answer is

2x - 4

. The choice

8x - 4

comes from wrongly adding

3

and

5

to get

8x

, and

-2x + 20

comes from reversing both subtractions.

Simplify

7a + 4b - a - 9b + 3

and explain how you know which terms can be combined.

Answer: $6a - 5b + 3$

Terms can be combined only when their variable parts match exactly. The

a

terms are

7a

and

-a

; since

-a

has coefficient

-1

7 - 1 = 6

, giving

6a

. The

b

terms are

4b

and

-9b

, and

4 - 9 = -5

, giving

-5b

. The constant

3

has nothing to pair with, so it stays. The

a

terms and

b

terms cannot merge because

a

and

b

are different quantities. Result:

6a - 5b + 3

. A check at

a = 1

b = 1

gives

7 + 4 - 1 - 9 + 3 = 4

for the original and

6 - 5 + 3 = 4

for the simplified form.

Mia says

6m + 4

simplifies to

10m

. Is she right? Explain what went wrong, and give a value of

m

that proves your point.

Answer: She is wrong; $6m + 4$ is already simplified because $6m$ and $4$ are not like terms.

6m

has variable part

m

and

4

has no variable, so they are not like terms and cannot be combined. Mia added the numbers and attached the variable, which changes the value of the expression. Test

m = 2

: the original gives

6(2) + 4 = 16

, but

10m

gives

10(2) = 20

. Since

16 \neq 20

, the expressions are not equivalent. Some expressions are already in simplest form, and

6m + 4

is one of them.

FAQ

Can I combine $5x$ and $5y$ since the coefficients are the same?: No. Like terms are decided by the variable part, not the coefficient. $5x$ and $5y$ represent different quantities, so $5x + 5y$ stays as it is. Matching coefficients never make terms combinable.
What happens to the variable when the coefficients add to zero?: The whole term disappears. In $7x - 7x + 3$ , the variable terms give $0x$ , and $0$ times anything is $0$ , so the expression simplifies to just $3$ . Do not carry the $x$ into the constant — writing $3x$ would be a different expression entirely.
Does the order of my final terms matter?: Not for correctness. $4 + 2x$ and $2x + 4$ are the same expression. Most teachers prefer variable terms first, in alphabetical order, with the constant at the end, because it makes answers easy to compare and easy to check.
How can I be sure my simplified expression is right?: Substitute a number for the variable into both the original and your simplified version. If both give the same value, that is strong evidence they are equivalent. Try a second number, ideally a negative one, since sign errors often hide when you only test positives.

Learn this with a teacher, not a page

The Crimsora tutor teaches Combining Like Terms live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

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