Like and unlike terms — simplification: Worked examples

This page is the worked-example gym for Like and unlike terms — simplification. The parent note told you the rule. Here we hit that rule from every possible angle — every sign, every "trap" input, a word problem, and an exam twist — so that no simplification problem can ever surprise you.

Before we start, the one idea we lean on everywhere. Picture terms as labelled buckets. A "bucket" is a variable part like $x$ , or $x^{2}$ , or $ab$ . You may only pour the count (the coefficient) from one bucket into another bucket with the exact same label. Different label = different bucket = keep separate.

The figure below draws this: three chalk buckets side by side. Two are labelled $x$ (chalk blue) with counts $3$ and $5$ — a yellow arrow pours them together into $8$ — while a third bucket labelled $y$ (chalk pink) stands apart with a pink arrow saying "keep separate". Read it as the visual definition of "like" (same label, combine) versus "unlike" (different label, leave alone) that every example below relies on.

Figure — Like and unlike terms — simplification

The scenario matrix

Every problem this topic can throw at you lands in one of these cells. The examples below are tagged with the cell(s) they cover.

Cell	Case class	What's tricky	Example
A	All-positive, one bucket	warm-up: pure counting	Ex 1
B	Mixed signs, one bucket	the minus belongs to the term	Ex 2
C	Multiple buckets, multiple powers	grouping $x^{2}$ vs $x$ vs constant	Ex 3
D	Multi-variable buckets	$ab$ vs $a$ vs $b$ vs $ba$	Ex 4
E	Result is zero (degenerate)	a bucket empties out	Ex 5
F	Hidden coefficient $1$ and lone $x$	invisible $1$ , invisible $+$	Ex 6
G	Real-world word problem	translate words → buckets	Ex 7
H	Exam twist: nested brackets + signs	distribute first, THEN combine	Ex 8

Cells to notice especially: E (an answer of $0$ — the bucket empties, and beginners panic), and D (where $ab$ and $ba$ look different but are the same bucket).

The examples

Worked example Example 1 — Cell A: all-positive, one bucket

Simplify $4 x + 6 x + x$ .

Forecast: guess the single-term answer before reading on. (How many $x$ 's in total?)

Confirm one bucket. Every term has the variable part $x$ . Why this step? We can only pour counts between identical labels; check the labels match first.
Read the hidden $1$ . The bare $x$ means $1 x$ — a bucket always carries a count, even when it's unwritten. Why this step? Forgetting the invisible $1$ is the most common off-by-one slip.
Add the counts: $4 + 6 + 1 = 11$ . Why this step? Distributive property: $4 x + 6 x + 1 x = (4 + 6 + 1) x$ .
Keep the label: answer $= 11 x$ . Why this step? We counted how many $x$ 's; the thing we're counting ( $x$ ) is unchanged, so its label rides along untouched.

Verify: put $x = 2$ . Original: $4 (2) + 6 (2) + 2 = 8 + 12 + 2 = 22$ . Answer: $11 (2) = 22$ . ✓ Same number, so the simplification preserved the value.

Worked example Example 2 — Cell B: mixed signs, one bucket

Simplify $9 y - 12 y + 5 y - 4 y$ .

Forecast: will the answer be positive or negative? Guess the sign first.

Attach each sign to its term. Rewrite as $(+ 9 y) + (- 12 y) + (+ 5 y) + (- 4 y)$ . Why this step? A minus sign is part of the coefficient, not a separate floating symbol. Making it explicit stops sign-dropping.
One bucket check: all four are $y$ . Good.
Add the signed counts: $9 - 12 + 5 - 4$ . Go left to right: $9 - 12 = - 3$ , then $- 3 + 5 = 2$ , then $2 - 4 = - 2$ . Why this step? Distributive property works with negatives too: $(9 - 12 + 5 - 4) y$ .
Answer: $- 2 y$ . Why this step? The count came out negative, so the surviving bucket holds a negative amount of $y$ ; the label $y$ is still unchanged.

Verify: put $y = 3$ . Original: $27 - 36 + 15 - 12 = - 6$ . Answer: $- 2 (3) = - 6$ . ✓

Worked example Example 3 — Cell C: several powers and a constant

Simplify $5 x^{2} - 8 x + 3 - 2 x^{2} + 11 x - 7$ .

Forecast: how many terms will survive in the final answer — 1, 2, or 3?

The figure below shows three chalk shelves stacked vertically — an $x^{2}$ shelf (blue), an $x$ shelf (yellow), and a constant shelf (pink) — each holding only its own terms and a yellow arrow pointing to the final descending-power answer. Use it to see that terms never jump shelves.

Sort into buckets by label:
- $x^{2}$ bucket: $5 x^{2}, - 2 x^{2}$
- $x$ bucket: $- 8 x, + 11 x$
- constant bucket (label = "no variable"): $+ 3, - 7$ Why this step? $x^{2}$ , $x$ , and a plain number are three different labels — the three shelves in the figure — so sorting first stops us illegally mixing them.
Combine inside each bucket:
- $x^{2}$ : $5 - 2 = 3 \Rightarrow 3 x^{2}$
- $x$ : $- 8 + 11 = 3 \Rightarrow 3 x$
- constant: $3 - 7 = - 4$ Why this step? Counting stays within a bucket; nothing crosses shelves.
Write in descending powers: $3 x^{2} + 3 x - 4$ . Why this step? Standard form makes later work (factoring, polynomials) predictable.

Verify: put $x = 2$ . Original: $5 (4) - 8 (2) + 3 - 2 (4) + 11 (2) - 7 = 20 - 16 + 3 - 8 + 22 - 7 = 14$ . Answer: $3 (4) + 3 (2) - 4 = 12 + 6 - 4 = 14$ . ✓

Worked example Example 4 — Cell D: multi-variable buckets (the

$ab$ = $ba$ trap) Simplify $3 ab + 2 ba - 5 a + a + 4 b$ .

Forecast: is $ab$ the same bucket as $ba$ ? Decide before step 1.

Normalise labels. Multiplication order doesn't matter: $ba = ab$ . So rewrite $2 ba$ as $2 ab$ . Why this step? $a \cdot b = b \cdot a$ (numbers/variables multiply in any order). Same label ⇒ same bucket. This is the whole trick of cell D.
Sort into buckets:
- $ab$ bucket: $3 ab, 2 ab$
- $a$ bucket: $- 5 a, + a$ (that lone $a$ is $+ 1 a$ )
- $b$ bucket: $+ 4 b$ Why this step? $ab$ , $a$ , and $b$ are three different labels (a product of two letters is not the same as either letter alone), so each must be counted on its own.
Combine:
- $ab$ : $3 + 2 = 5 \Rightarrow 5 ab$
- $a$ : $- 5 + 1 = - 4 \Rightarrow - 4 a$
- $b$ : $4 b$ (alone) Why this step? We add signed counts only within each label; the $b$ bucket has just one term, so nothing to combine — it stays as is.
Answer: $5 ab - 4 a + 4 b$ . Why this step? Three different labels survived, so they stay as three separate terms — unlike terms can never be merged.

Verify: put $a = 2, b = 3$ . Original: $3 (6) + 2 (6) - 5 (2) + 2 + 4 (3) = 18 + 12 - 10 + 2 + 12 = 34$ . Answer: $5 (6) - 4 (2) + 4 (3) = 30 - 8 + 12 = 34$ . ✓

Worked example Example 5 — Cell E: a bucket empties to zero (degenerate)

Simplify $7 m + 4 - 7 m$ .

Forecast: does the answer still contain $m$ ? Yes or no — commit now.

Sort buckets: $m$ bucket: $7 m, - 7 m$ . Constant bucket: $+ 4$ . Why this step? $m$ and a plain number are different labels; sorting first shows us exactly which terms are eligible to combine.
Combine the $m$ bucket: $7 - 7 = 0$ , so the $m$ term is $0 \cdot m = 0$ . Why this step? When counts cancel, the bucket becomes empty. Zero of anything is just zero — the label disappears with its count.
Drop the zero term. $0 + 4 = 4$ . Why this step? Writing " $0 m$ " is legal but pointless; a zero term contributes nothing, so we omit it.
Answer: $4$ (a pure constant — no variable left!). Why this step? Only the constant bucket had a surviving count, so the whole expression collapses to that single number.

Verify: put $m = 100$ . Original: $700 + 4 - 700 = 4$ . Answer: $4$ . ✓ The value doesn't depend on $m$ at all, which is exactly why the $m$ vanished.

Worked example Example 7 — Cell G: real-world word problem

A shop sells notebooks at $\$ n $e a c han d p e n s a t$ $p $e a c h . O n M o n d a y A i s hab u y s * * 5 n o t e b oo k s an d 3 p e n s * *; o n T u es d a y s h e b u y s * * 2 n o t e b oo k s an d 8 p e n s * *; t h e n s h e * * r e t u r n s 1 n o t e b oo k * * . W r i t e h er t o t a l s p e n d a s a s im pl i f i e d e x p r ess i o n, t h e n f in d i tw h e nan o t e b oo k cos t s$ $4 $an d a p e n cos t s$ $2$.

Forecast: guess how many notebooks (net) and pens she paid for.

Translate each purchase into buckets. Notebooks live in the $n$ bucket, pens in the $p$ bucket: $5 n + 3 p + 2 n + 8 p - 1 n$ Why this step? Money-per-notebook is a different "currency" from money-per-pen — exactly the like/unlike distinction. A return is a negative count.
Combine each bucket:
- $n$ : $5 + 2 - 1 = 6 \Rightarrow 6 n$
- $p$ : $3 + 8 = 11 \Rightarrow 11 p$ Why this step? We can only total notebooks with notebooks and pens with pens; adding the signed counts within each currency gives the net quantity of each item.
Simplified total: $6 n + 11 p$ dollars. Why this step? $n$ and $p$ are unlike terms (different prices), so they cannot merge — the simplest honest form keeps both, each with its net count.
Plug in $n = 4, p = 2$ : $6 (4) + 11 (2) = 24 + 22 = 46$ . Why this step? The letters were stand-ins for the unknown prices; substituting the actual prices turns the general formula into the specific dollar amount she paid.

Verify (units + arithmetic): dollars/notebook × notebooks + dollars/pen × pens = dollars ✓. Direct count: $6$ notebooks at $4 = $24; $11$ pens at $2 = $22; total $46. ✓ Matches.

Worked example Example 8 — Cell H: exam twist, brackets first

Simplify $2 (3 x - 4) - 3 (x - 5) + x$ .

Forecast: the brackets hide like terms. Guess the final coefficient of $x$ .

Distribute — remove brackets before combining. Multiply each bracket by its outside number, sign included:
- $2 (3 x - 4) = 6 x - 8$
- $- 3 (x - 5) = - 3 x + 15$ ← the $- 3$ multiplies both inside terms; $- 3 \times - 5 = + 15$ . Why this step? You can't combine terms trapped inside brackets. The distributive property frees them. The most-missed part is $- 3 \times - 5 = + 15$ : a negative times a negative is positive.
Rewrite fully expanded: $6 x - 8 - 3 x + 15 + x$ . Why this step? Laying all terms out flat, with no brackets, is what lets us finally see which terms share a label and are eligible to combine.
Sort buckets:
- $x$ bucket: $6 x, - 3 x, + 1 x$
- constant bucket: $- 8, + 15$ Why this step? $x$ -terms and plain numbers are different labels; grouping them makes the two independent additions visible before we do them.
Combine:
- $x$ : $6 - 3 + 1 = 4 \Rightarrow 4 x$
- constant: $- 8 + 15 = 7$ Why this step? We add signed counts only within each label, giving one net $x$ -term and one net constant.
Answer: $4 x + 7$ . Why this step? Two different labels survived ( $x$ and constant), so they stay as two separate terms — unlike terms cannot be merged into one.

Verify: put $x = 3$ . Original: $2 (9 - 4) - 3 (3 - 5) + 3 = 2 (5) - 3 (- 2) + 3 = 10 + 6 + 3 = 19$ . Answer: $4 (3) + 7 = 19$ . ✓

Recall

Recall When a variable cancels to zero, what does the answer contain?

Only the remaining constant (or other surviving buckets). The cancelled variable disappears entirely. A vanished variable means an error ::: False — it means two equal-and-opposite terms cancelled, which is completely valid.

Recall Is

$ab$ the same bucket as $ba$ ? Yes — multiplication order doesn't matter, so $ab = ba$ ; combine them.

Recall In a bracket like

$- 3 (x - 5)$ , what does the $- 3$ multiply? Both terms inside: $- 3 \cdot x = - 3 x$ and $- 3 \cdot (- 5) = + 15$ .

Connections

Like and unlike terms — simplification — the parent rule these examples drill
2.1.01-Variables-constants-and-coefficients — reading the hidden coefficient $1$ (Ex 1, 4, 6)
2.2.01-Distributive-property — the engine behind combining and behind Ex 8's brackets
2.1.03-Addition-and-subtraction-of-algebraic-expressions — these examples are the core of that skill
3.1.01-Solving-linear-equations-one-variable — cancellation (Ex 5) is what lets you solve
4.1.01-Polynomials-introduction — Ex 3's descending-power form is polynomial standard form