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Adding & Subtracting Like Terms

Maths · Y8 Algebra
Year 8
Lesson Guide
Explanation
You can only add apples to apples. LIKE terms have exactly the same variable part — same letters raised to the same powers: 3x, −7x and x/2 are like; 3x, 3x2 and 3xy are all different species. The coefficient is allowed to differ — that's the part we'll be adding. One subtlety: 5ab and −2ba ARE like terms, because ab = ba (multiplication order doesn't matter). But x and x2 are as unlike as apples and apple crates.
Your Turn
Pick the like terms from: 4m, 3mn, −m, 2m2, 7nm. Check: 4m and −m are like; 3mn and 7nm are like (a second pair).
Watch Out
Powers must match exactly: 2x and 2x2 are NOT like terms. The exponent is part of the species.
Explanation
Collecting like terms: group each species, add the coefficients. 5x + 3 + 2x − 1: the x-terms give 5x + 2x = 7x; the constants give 3 − 1 = 2. Answer: 7x + 2 — and it STOPS there. 7x + 2 is not 9x: an x-term and a constant are different species and can never merge. Check with a number: at x = 10, the original gives 50+3+20−1 = 72, and 7x + 2 = 72. Simplifying never changes the value — it just shortens the story.
Your Turn
Simplify 8a + 5 − 3a + 2. Check: 5a + 7.
Watch Out
7x + 2 ≠ 9x. Unlike terms sit side by side forever — the + sign between them is the final answer.
Explanation
Negatives are where marks are lost. The fix: every term OWNS the sign in front of it. Simplify 6y − 4 − 9y + 7 by reading it as (+6y) (−4) (−9y) (+7). Now collect: y-terms +6y − 9y = −3y (Chapter 1 integer rules: 6 − 9 = −3); constants −4 + 7 = +3. Answer: −3y + 3. When you rearrange terms, the sign moves WITH the term like a shadow — drop it and the whole answer flips.
Your Turn
Simplify 3p − 8 − 7p + 2. Check: −4p − 6.
Watch Out
Moving a term without its sign is the classic error: in 6y − 9y, the 9y carries the minus. Underline terms with signs before collecting.
Explanation
With several variables, sort into species first. 4a + 7b − a + 2b − 3: the a-terms give 4a − a = 3a (remember −a is −1a, so 4 − 1 = 3); the b-terms give 7b + 2b = 9b; the constant −3 has no partner and stays. Answer: 3a + 9b − 3. Resist the urge to 'finish' it — 3a + 9b is NOT 12ab. An expression with three different species in it is already fully simplified.
Your Turn
Simplify 5x + 2y − 3x + y − 4 + 6. Check: 2x + 3y + 2.
Watch Out
−a has coefficient −1, not 0 and not −. Writing 4a − a = 4 (forgetting the a survives) is a classic slip.
Explanation
The full sorting challenge: 3x2 + 5x − x2 + 2xy + 4x − yx has THREE species. The x2 tribe: 3x2 − x2 = 2x2. The x tribe: 5x + 4x = 9x. The xy tribe: 2xy − yx = 1xy (since yx = xy). Answer: 2x2 + 9x + xy. Two traps live here: x2 and x never mix (different powers), and subtracting a whole term leaves the species behind — 3x2 − x2 = 2x2, not 2 or 3.
Your Turn
Simplify 6ab + 2a2 − ba + 3a2. Check: 5ab + 5a2.
Watch Out
3x2 − x2 = 2x2. The x2 doesn't vanish — you subtracted ONE of them, not all of them.
Explanation
Like terms in the wild: a rectangle with length 2x + 3 and width x has perimeter (2x+3) + (2x+3) + x + x = 6x + 6 — one tidy formula for every possible x. The full checklist: underline each term with its sign; sort into species; add the coefficients within each species; then sanity-check with a substitution (x = 5: original 36, and 6(5) + 6 = 36 ✓). That last habit catches almost every collecting error. Quiz and exercises!
Your Turn
A triangle has sides 3a, 2a + 4 and a − 1. Perimeter? Check: 6a + 3.
Watch Out
In word problems write ALL the pieces down before collecting — the fastest way to lose a term is to hold it in your head.
📌 Key Points — this lesson in a nutshell
  • Like terms have exactly the same variable part — same letters, same powers (ab = ba counts as like).
  • Add or subtract the coefficients only; the variable part is unchanged: 5x + 2x = 7x.
  • Unlike terms never merge: 7x + 2 stays 7x + 2, and 3a + 9b is NOT 12ab.
  • The sign in front travels with its term: 6y − 9y = −3y.
  • −a means −1a, and 3x2 − x2 = 2x2 — the species survives subtraction.
  • Check any simplification by substituting a number into both versions.
Step 1: Introduction
Step 1 of 6
Apples with apples LIKE terms 3x, −7x, x/2 same variable part: x NOT like 3x, 3x2, 3xy, 3 different variable parts Like terms have EXACTLY the same variable part — same letters, same powers. Coefficients can differ. 5ab and −2ba ARE like (ab = ba, order doesn't matter). x and x2 are as different as apples and apple crates.
Simplify 5x + 3 + 2x − 1 5x + 3 + 2x − 1 5x + 2x = 7x 3 − 1 = 2 = 7x + 2
Simplify 6y − 4 − 9y + 7 Underline each term WITH its sign: +6y −4 −9y +7 y-terms: +6y − 9y = −3y constants: −4 + 7 = +3 = −3y + 3 Integer rules from Chapter 1 are doing all the work here: 6 − 9 = −3.
Simplify 4a + 7b − a + 2b − 3 a-terms 4a − a = 3a b-terms 7b + 2b = 9b constants −3 stays = 3a + 9b − 3 Remember: −a means −1a. And 3a + 9b does NOT become 12ab — different species never merge, no matter how tidy it looks.
Simplify 3x2 + 5x − x2 + 2xy + 4x − yx x2 species 3x2 − x2 = 2x2 x species 5x + 4x = 9x xy species 2xy − yx = xy = 2x2 + 9x + xy yx = xy (order doesn't matter) and x2 is its OWN species — three tribes in, three tribes out. 2x2 − x2 = x2 (not 2).
Algebra measures the fence 2x + 3 x Perimeter = 2 lengths + 2 widths = (2x+3) + (2x+3) + x + x = 6x + 6 check x = 5: 13+13+5+5 = 36 ✓ The like-terms checklist: 1. Underline each term with its sign. 2. Sort into species. 3. Add coefficients per species. 4. Sanity-check by substituting.
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