Collecting Like Terms
Why is $3x + 2x = 5x$ but $3x + 2y$ stays as it is? Learn the secret of "like terms" and how to tidy up messy expressions into clean, simple form.
Printable Worksheets
Print or save as PDF, or build a custom worksheet from any module's questions.
Simplify $3a + 2a + 5b - b$. Explain why you can or cannot combine each pair of terms.
Collecting like terms is the algebra version of tidying your room. You group similar things together. In algebra, like terms are terms that have exactly the same variable part.
$3x$ and $5x$ are like terms because both have $x$. We add their coefficients: $3 + 5 = 8$, so $3x + 5x = 8x$. But $3x$ and $5y$ are unlikedifferent variables, so they cannot be combined.
Know
- What like terms are
- That only like terms can be combined
- That coefficients are added/subtracted
Understand
- Why $x$ and $y$ cannot be combined
- That the variable part stays unchanged
- Why signs must be tracked carefully
Can Do
- Identify like and unlike terms
- Simplify expressions by collecting like terms
- Handle negative coefficients
Wrong: $3a + 2b = 5ab$
Right: $3a + 2b$ cannot be simplified. Different variables = different categories.
Wrong: $4x - 7x = -3$ (forgetting the $x$)
Right: $4x - 7x = -3x$. The variable stays! Only the coefficient changes.
Think of collecting like terms as sorting items into bins. All the $x$ terms go in one bin, all the $y$ terms in another, and all the numbers in a third bin.
Take $3x + 2y + 5x - y + 7$. Sort: $x$-terms are $3x$ and $5x$, $y$-terms are $2y$ and $-y$, constants are $+7$. Then combine each bin: $8x + y + 7$.
Two terms are "like" only if they have the exact same variable part same letters, same powers. $3x^2$ and $5x^2$ are like. $3x^2$ and $5x$ are not like (different powers).
$2ab$ and $5ab$ are like (both have $ab$). $2ab$ and $3a$ are unlike (one has $ab$, one has just $a$). Even if letters overlap, the whole variable part must match.
| Terms | Variable Part | Like? | Result |
|---|---|---|---|
| $4a$, $7a$ | both $a$ | Yes | $11a$ |
| $2xy$, $5xy$ | both $xy$ | Yes | $7xy$ |
| $3x$, $3y$ | $x$ vs $y$ | No | stays $3x + 3y$ |
| $2x^2$, $3x$ | $x^2$ vs $x$ | No | stays $2x^2 + 3x$ |
| $5$, $-2$ | no variable | Yes | $3$ |
When terms have negative coefficients, the minus sign belongs to the term. $4a - 7a$ means $4a + (-7a) = -3a$. Treat it like adding signed numbers.
Look at $5x - 8x + 2x$. Think: $5 - 8 + 2$. $5 - 8 = -3$, then $-3 + 2 = -1$. So the answer is $-x$ (or $-1x$). Work left to right, tracking the sign.
Here is the complete method for simplifying any expression: (1) Identify all terms, (2) Group like terms, (3) Combine coefficients, (4) Write the simplified answer.
Watch Me Solve It · Worked example
- 1Identify all terms$4a$, $+3b$, $-7a$, $+2$, $-2b$, $+5$Each term separated by + or -. The sign stays attached to what follows it.
- 2Group like terms together$4a - 7a + 3b - 2b + 2 + 5$$a$-terms first: $4a$ and $-7a$. $b$-terms: $+3b$ and $-2b$. Constants: $+2$ and $+5$.
- 3Combine $a$-terms$4a - 7a = (4 - 7)a = -3a$Four $a$'s minus seven $a$'s = three $a$'s owing. The coefficient becomes $-3$.
- 4Combine remaining terms$3b - 2b = b$ and $2 + 5 = 7$Three $b$'s minus two $b$'s = one $b$ (just $b$). The numbers add to $7$.
- 5Write the final simplified expression$-3a + b + 7$Usually written with the positive term first: $b - 3a + 7$ is also fine. Check: count original terms (6) vs final (3), simpler!
Like Terms Rule
- Same variable + same power = like
- Add/subtract the coefficients only
- Variable part never changes
- Constants combine with constants
Unlike Terms
- Different variables = unlike
- Different powers = unlike
- Leave them as they are
- $3x + 2y$ stays $3x + 2y$
Method
- Step 1: Identify all terms
- Step 2: Group like terms
- Step 3: Combine coefficients
- Step 4: Write simplified answer
Sign Tips
- $4a - 7a = -3a$ (not $-3$)
- $-x$ means $-1x$
- $5x - 8x + 2x = -x$
- Track signs carefully!
How are you completing this lesson?
Brain Trainer · 4 problems
Four problems. Work each one, then reveal the answer.
1 Simplify $3x + 7x$.
Both have $x$. $3 + 7 = 10$. $10x$$10x$2 Simplify $4a + 2b - 3a + 5b$.
$a$-terms: $4a - 3a = a$. $b$-terms: $2b + 5b = 7b$. $a + 7b$$a + 7b$3 Simplify $5m - 8m + 2m$.
$5 - 8 + 2 = -1$. $-m$ (or $-1m$).$-m$4 Simplify $2x + 3y - x + 4 - 2y + 6$.
$x$-terms: $2x - x = x$. $y$-terms: $3y - 2y = y$. Constants: $4 + 6 = 10$. $x + y + 10$$x + y + 10$
Show Your Working · 3 questions
Q6. Simplify $6x + 3y - 2x + y$.
$x$-terms: $6x - 2x = 4x$ (1). $y$-terms: $3y + y = 4y$ (1). $4x + 4y$.
Q7. Simplify $3a - 5b + 2a + 7 - b - 4$.
$a$-terms: $3a + 2a = 5a$ (1). $b$-terms: $-5b - b = -6b$ (1). Constants: $7 - 4 = 3$ (1). $5a - 6b + 3$.
Q8. A rectangle has sides $(3x + 2)$ and $(5x - 1)$. Write and simplify an expression for its perimeter.
Perimeter = $2(3x+2) + 2(5x-1)$ (1) $= 6x + 4 + 10x - 2$ (1) $= 6x + 10x + 4 - 2$ (1) $=$ $16x + 2$ (1).
Extension Problems
Ready for a bigger challenge? Try these extension problems.
Key Concept
Review the main ideas from this lesson.
Formulas
Key formulas and rules.
Watch Out
Common mistakes to avoid.
Check
Always verify your answers.
Practice
Keep practicing to master.
Next
Build on these skills.
Interactive: Algebra Machine
Substitute numbers into algebraic expressions and see them evaluate step by step.
Your Badges
0 of 6Mark lesson as complete
Tick when you've finished Learn, Practice and the Stretch. Earns +90 XP and +25 coins.