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.
Today's hook: You have 3 apples and 2 apples. You have 5 apples. But 3 apples and 2 bananas? You just have 3 apples and 2 bananas. That's collecting like terms.
0/5QUESTS
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Simplify $3a + 2a + 5b - b$. Explain why you can or cannot combine each pair of terms.
Record in workbook.
1
The Big Idea
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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 unlike — different variables, so they cannot be combined.
$3x$ and $7x$ are like terms. $3x$ and $3y$ are not. The variable must match exactly.
Add coefficients only
$4a + 2a = 6a$. The variable stays the same. Only the numbers in front change.
Signs matter
$4a - 7a = -3a$. The minus sign belongs to the 7, so it's $4 + (-7) = -3$.
2
What You'll Master
objectives
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
3
Words You Need
vocabulary
Like TermsTerms with the exact same variable part. e.g. $3x$ and $7x$, or $2ab$ and $5ab$.
Unlike TermsTerms with different variables. e.g. $3x$ and $5y$. Cannot be combined.
SimplifyTo make an expression shorter by collecting like terms. $3x+5x$ simplifies to $8x$.
CoefficientThe number in front of a term. When collecting, only coefficients are added/subtracted.
Constant TermA term with no variable, like $+5$ or $-3$. Constants combine with constants.
Algebraic SumThe result of adding/subtracting terms. e.g. $4x - 7x = -3x$.
4
Spot the Trap
heads-up
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.
5
The Sorting Game
+5 XP
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: $8x + y + 7$.
$3x + 2y + 5x - y + 7 = 8x + y + 7$
Highlight or circle
In your book, use different colours for each type of term before combining.
Rearrange first
$3x + 2y + 5x$ → $3x + 5x + 2y$. Group like terms together before adding.
Constants too
$5 + 3 - 2 = 6$. Numbers without variables are like terms with each other.
6
What Counts as "Like"?
+5 XP
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.
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.
$5x - 8x + 2x = -x$
Think of the signs
$4a - 7a$ = four $a$'s minus seven $a$'s = three $a$'s in debt = $-3a$.
Use a number line
$5 - 8$: start at 5, go 8 left. You land on $-3$. Visualise it!
$-x = -1x$
When the coefficient becomes $-1$, just write $-x$. Don't write the 1.
8
The Full Process
+5 XP
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
Watch Me Solve It · Full simplification
+15 XP per step
Q
PROBLEM
Simplify $4a + 3b - 7a + 2 - 2b + 5$.
1
Identify all terms
$4a$, $+3b$, $-7a$, $+2$, $-2b$, $+5$
Each term separated by + or -. The sign stays attached to what follows it.
2
Group like terms together
$4a - 7a + 3b - 2b + 2 + 5$
$a$-terms first: $4a$ and $-7a$. $b$-terms: $+3b$ and $-2b$. Constants: $+2$ and $+5$.
3
Combine $a$-terms
$4a - 7a = (4 - 7)a = -3a$
Four $a$'s minus seven $a$'s = three $a$'s owing. The coefficient becomes $-3$.
4
Combine remaining terms
$3b - 2b = b$ and $2 + 5 = 7$
Three $b$'s minus two $b$'s = one $b$ (just $b$). The numbers add to $7$.
5
Write 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!
Nice work — XP earned
Answer$-3a + b + 7$
9
Common Pitfalls
heads-up
Combining unlike terms
$3x + 2y = 5xy$ is wrong. Only like terms combine. $3x + 2y$ cannot be simplified at all.
Fix: check that the variable part is identical before combining. If different, leave it alone.
Dropping the variable
$4x - 7x = -3$ is wrong — the $x$ disappeared!
Fix: the variable stays forever. $4x - 7x = -3x$. Only the coefficient changes.
Ignoring signs when combining
$5x - 8x = 3x$ is wrong. The minus belongs to the 8. It's $5 + (-8) = -3$.
Fix: think of each term with its sign. $5x - 8x = (5 - 8)x = -3x$.
Copy Into Your Books
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!
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