POP

G2.6 Simplifying Algebraic Expressions

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01 — Explanation

What are "like terms", and why can I only combine those?

Like terms share the exact same variable part — same letter(s), same power. $3a$ and $7a$ are like terms; so are $5x^2$ and $-2x^2$ ; so are $4pq$ and $pq$ . But $3a$ and $4b$ are not like terms, and neither are $5x$ and $5x^2$ — one is a "length" ( $x$ ) and the other an "area" ( $x^2$ ), and you can't add those together any more than you can add 3 apples to 4 bananas.

02 — Explanation

How do I collect like terms systematically?

Use a three-step method: Identify each group of like terms (circle or box them, keeping the sign attached to the term that follows it), group them next to each other, then combine the coefficients.

Example: simplify $7x + 2y - 4x - 5y$ .

Identify: $x$ -terms are $(+7x), (-4x)$ ; $y$ -terms are $(+2y), (-5y)$ .
Group: $7x - 4x + 2y - 5y$ .
Combine: $7-4=3 \Rightarrow 3x$ ; and $2-5=-3 \Rightarrow -3y$ .

Answer: $3x - 3y$ .

Common Mistake

writing $8x - 2x = 10x$ because the minus sign gets mentally detached from the term it belongs to. Always treat the sign as part of the term: here the terms are $+8x$ and $-2x$ , giving $6x$ .

03 — Explanation

What happens when powers or several variables are involved?

The rule doesn't change — only terms with identical variable and power combine.

Example: simplify $2a^2 + 5b - 2a - 3b + 5a^2$ .

$a^2$ terms: $2+5=7 \Rightarrow 7a^2$
$b$ terms: $5-3=2 \Rightarrow 2b$
the lone $-2a$ term has no partner, so it stays as is.

Answer: $7a^2 + 2b - 2a$ .

Exam Tip

a variable written alone has an invisible coefficient of 1 — so $b$ is $1b$ and $-x$ is $-1x$ . Forgetting this causes errors like simplifying $5x - x$ to $5$ instead of $4x$ . It's also possible for a group to vanish entirely: in $ab + bc + 2ab - bc$ , the $bc$ terms give $1-1=0$ , so they disappear from the final answer.

04 — Worked example

Worked exam-style question

Simplify: (a) $p+p+p+p$ (b) $8e+9f-2f+4e$ (c) $4\times3\times y$

(a) Four lots of $p$ added: $4p$ .

(b) Group: $e$ -terms $8+4=12$ ; $f$ -terms $9-2=7$ . Answer: $12e+7f$ .

(c) $4\times3=12$ , so the answer is $12y$ .

Exam Tip

on a question like (b), partial credit is often available for correctly simplifying just one of the two term-types, so always attempt every group even under time pressure.

05 — Explanation

What if the terms need multiplying together instead of collecting?

Simplifying isn't always about adding like terms — sometimes two separate terms are being multiplied, and the rule is completely different: multiply the numbers together, then multiply the letters together.

Example: simplify $3a \times 4b$ and $2x \times 3x$ .

$3a \times 4b$ : multiply numbers $3\times4=12$ , multiply letters $a\times b=ab$ . Answer: $12ab$ .
$2x \times 3x$ : multiply numbers $2\times3=6$ , and $x\times x=x^2$ (not $2x$ ). Answer: $6x^2$ .

Common Mistake

writing $x\times x=2x$ instead of $x^2$ . Multiplying a letter by itself always produces a power, never a coefficient of 2 — that confusion comes from mixing up the rules for addition ( $x+x=2x$ ) and multiplication ( $x\times x=x^2$ ).

06 — Explanation

How do I check I haven't left the expression half-simplified?

Before moving on, scan the final line and ask: "are there still two terms with the same letter and power?" If so, the expression isn't finished. Writing $2x+4+3x$ as a final answer, instead of combining the $x$ -terms to get $5x+4$ , is a frequent way to lose an easy mark despite having done all the hard work correctly.