Collecting like terms and tidying expressions — built up one small step at a time.
6 lessons · 35 questions (A–G) · 20 calculation + 15 word problems · no time pressure
Simplifying in algebra means tidying an expression by collecting its like terms — the parts that count the same thing — so a longer expression becomes a shorter one worth exactly the same. Pupils most often slip by combining terms that are not alike (adding an x to a plain number, or an x to an x2), or by dropping a minus sign when a term is being subtracted.
At Improve Tuition, a qualified teacher pinpoints exactly which terms a pupil is mismatching and rebuilds the method in small, worked steps — with read-aloud, comfort spacing and a reading tint for pupils who take in maths more easily that way.
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Lesson 1
Like terms
Aim: Recognise like terms — the parts of an expression that can be combined.
In algebra a letter stands for an amount we do not know yet. A term is a single part of an expression, such as 5m or 7. Like terms use exactly the same letter, so they count the same kind of thing — you can add 5m and 2m just as you would add 5 apples and 2 apples. Plain numbers are like terms with one another. Terms with different letters cannot be combined.
Like terms use the same letter: 4m and 2m can be combined; the number 3 stays on its own.
Worked example
In the expression 4m + 3 + 2m, which terms are like terms?
Look at each term in turn: 4m, then 3, then 2m.
4m and 2m both use the letter m, so they are like terms — the same kind of thing.
The 3 is just a number, so it is a different kind of term on its own.
So the like terms are 4m and 2m; the 3 stays separate.
Check: only terms with the same letter can be grouped, and m matches m. ✓
Watch out
Different letters are different things: 4m and 4n cannot be combined, just as apples and oranges can’t go in one pile.
Try one
Which terms are like terms in 6p + 5q + p?
Show answer
Check the letters: 6p and p both use p; 5q uses q.
Aim: Collect like terms with a single letter by adding the number parts.
To simplify, add up the like terms. The number in front of a letter is its coefficient; to combine like terms you add the coefficients and keep the same letter. A letter on its own, like m, means 1m.
Add the coefficients and keep the letter: 5m + 2m = 7m.
Worked example
Simplify 5m + 2m.
Both terms use the letter m, so they are like terms.
Add the coefficients (the numbers in front): 5 + 2 = 7.
Keep the same letter, m, on the end.
So 5m + 2m = 7m.
Check: 5 lots of m plus 2 lots of m really is 7 lots of m. ✓
Watch out
Keep the letter the same — you add the coefficients, not the letters. 5m + 2m is 7m, never 10m or m².
Try one
Simplify 8w + w.
Show answer
w on its own means 1w, so this is 8w + 1w.
Add the coefficients: 8 + 1 = 9.
Keep the letter: 9w.
Check: 8 lots of w and one more w is 9 lots of w. ✓
Aim: Collect like terms that involve subtraction, remembering a lone letter means 1 of it.
Subtracting like terms works the same way — subtract the coefficients. A minus sign belongs to the term that follows it, so −m means −1m. The result can sometimes even be negative.
Subtracting removes tiles: from 7t take away 3t to leave 4t.
Worked example
Simplify 7t − 3t.
Both terms use t, so they are like terms.
Subtract the coefficients: 7 − 3 = 4.
Keep the letter t.
So 7t − 3t = 4t.
Check: take 3 lots of t away from 7 lots of t and 4 lots are left. ✓
Watch out
The minus sign sticks to its term. In 9t − t the second term is 1t, so the answer is 8t, not 9.
Try one
Simplify 4c − 9c.
Show answer
Both use c, so subtract the coefficients: 4 − 9.
4 − 9 = −5.
Keep the letter: −5c.
Check: owing 9 of something when you have only 4 leaves you 5 short, so −5c. ✓
Aim: Simplify expressions with different letters and numbers by grouping each kind separately.
When an expression mixes different letters and numbers, sort it into groups: all the terms in one letter together, all the terms in another letter together, and all the plain numbers together. Combine each group on its own, then write the results side by side.
Group each letter separately: the p-terms make 9p; the single q-term stays 2q.
Worked example
Simplify 6p + 2q + 3p.
Sort by kind: the p-terms are 6p and 3p; the q-term is 2q.
Combine the p-terms: 6p + 3p = 9p.
There is only one q-term, so 2q stays as it is.
Write the groups together: 9p + 2q.
Check: nothing is left over, and the p’s and q’s were kept apart. ✓
Watch out
Keep each letter in its own group. 9p + 2q cannot be made shorter — it is already simplified.
Try one
Simplify 5a + 4 + 2a + 3.
Show answer
Group the a-terms: 5a + 2a = 7a.
Group the numbers: 4 + 3 = 7.
Write them together: 7a + 7.
Check: the letters and the numbers were combined separately. ✓
Aim: Rearrange and simplify carefully, keeping each term’s sign with it.
You can reorder terms to bring like terms together, but each term must carry its own sign as it moves. A term written with a minus in front stays negative wherever you put it. Group the like terms, keeping their signs, then combine.
Join the like terms, keeping signs: the x-terms give 5x and the numbers give 3.
Worked example
Simplify 8x + 5 − 3x − 2.
Mark each term with its sign: +8x, +5, −3x, −2.
Bring like terms together, keeping signs: (8x − 3x) and (5 − 2).
Combine the x-terms: 8x − 3x = 5x.
Combine the numbers: 5 − 2 = 3.
So the answer is 5x + 3. Check: every term kept its own sign. ✓
Watch out
When you move a term, take its sign with it. The −3x must stay −3x, not become +3x.
Try one
Simplify 4y − 7 + y + 10.
Show answer
Signs: +4y, −7, +y, +10.
y-terms: 4y + y = 5y. Numbers: −7 + 10 = 3.
So the answer is 5y + 3.
Check: y on its own counted as 1y, and the signs were kept. ✓
Aim: Form an expression from a real situation, then simplify it.
Real problems often ask you to add several amounts written with letters — the sides of a shape, or items collected over time. Write down each part as a term, then collect the like terms to give one tidy expression. Write the expression first, simplify second.
Add all four sides: 4x + 3 + 4x + 3 collects to 8x + 6.
Worked example
A rectangle has two long sides of length 4x and two short sides of length 3. Write its perimeter, simplified.
The perimeter is the total of all four sides: 4x + 3 + 4x + 3.
Group the like terms: the x-terms 4x + 4x, and the numbers 3 + 3.
Combine: 4x + 4x = 8x, and 3 + 3 = 6.
So the perimeter is 8x + 6.
Check: two lots of 4x is 8x, and two lots of 3 is 6. ✓
Watch out
Add every side once. Write the full expression first, then simplify — don’t try to do both at once.
Try one
Mia has 3n stickers and 5 spare ones. She is given 2n more stickers. Write how many she has, simplified.
Show answer
Write the parts as terms: 3n + 5 + 2n.
Group the n-terms: 3n + 2n = 5n. The number 5 stays.
So she has 5n + 5.
Check: the sticker terms combined and the spare 5 stayed separate. ✓
31.A stall has 6c cans and 4d drinks. It sells 2c cans and d drinks. Write what is left, simplified.
Answer:
Show working
Start: 6c + 4d. Sell 2c cans and d drinks, so subtract: −2c − d.
Collect: 6c − 2c = 4c, and 4d − d = 3d.
So what is left is 4c + 3d. ✓
32.Anna has 5x stickers and 7 spare ones. She gives away 2x stickers and 3 spare ones. Write what she has left, simplified.
Answer:
Show working
Start: 5x + 7. Give away 2x and 3, so subtract: −2x − 3.
Collect: 5x − 2x = 3x, and 7 − 3 = 4.
So she has 3x + 4 left. ✓
33.A path’s first row uses a + b slabs, its second row 2a slabs, and its third row 3b slabs. Write the total number of slabs, simplified.
Answer:
Show working
Total = (a + b) + 2a + 3b.
Collect: a + 2a = 3a, and b + 3b = 4b.
So the total is 3a + 4b. ✓
34.A number machine starts at 0, adds 4m, then subtracts m, then adds 2m. Write the output, simplified.
Answer:
Show working
Output = 4m − m + 2m.
Collect: 4 − 1 + 2 = 5.
So the output is 5m. ✓
35.A rectangle has two long sides each of length (2x + 1) and two short sides each of length x. Write the perimeter, simplified.
Answer:
Show working
Perimeter = (2x + 1) + (2x + 1) + x + x.
Collect: 2x + 2x + x + x = 6x, and 1 + 1 = 2.
So the perimeter is 6x + 2. ✓
0 of 35 answered
Guide
What does simplifying algebra mean?
Simplifying algebra means rewriting an expression in its shortest form by collecting like terms — the parts that count the same thing — without changing its value. For example, 3a + 2a + 4 collects to 5a + 4, and 7x − 2x + y becomes 5x + y. The expression is shorter, but worth exactly the same for every value of the letters.
Like terms share the same letter part: 3a and 2a are like terms; 3a and 3b are not, and neither are x and x2. You can only add or subtract like terms — unlike terms simply sit side by side. ‘Reducing’ an expression means the same thing: keep collecting until nothing else will combine.
Common mistakes. Pupils most often combine terms that are not alike (turning x + x2 into a single term), drop a minus sign when a term is being subtracted, or stop too early and leave like terms uncollected.
Common questionsWhat are like terms?
Like terms have exactly the same letter part, such as 4y and 7y, or 2ab and 5ab. You add or subtract their number parts and keep the letter part the same.
Can you simplify x + x2?
No. x and x2 are not like terms because their letter parts are different, so x + x2 is already in its simplest form and cannot be combined.
How do you collect like terms?
Group the terms that share the same letter part, then add or subtract their numbers. For example, 6m + 2m − 3 collects to 8m − 3.
What is the difference between simplifying and solving?
Simplifying tidies an expression into a shorter, equal form and has no equals sign. Solving works out the value of the letter in an equation, which does have an equals sign.
Does simplifying change the value of an expression?
No. A simplified expression is equal to the original for every value of the letter; it is just written more briefly.
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