KS3 · Maths

Simplifying by collecting like terms

Three apples plus two apples is five apples. Three apples plus two bananas? Still just three apples and two bananas. Algebra works the same way.

Maths · Algebra

Same kind, or different kind?

Put each term in its group, then switch rounds and sort the very same terms again.

Ignore the number in front. Which letter or letters does each term use?

Still to sort

Just a (0)

The only letter is a (ignore the number in front).

Just x (0)

The only letter is x, whatever little power it wears.

a and b (0)

Both a and b appear.

d and e (0)

Both d and e appear.

b and c (0)

Both b and c appear.

13 of 13 still to sort.

Pick a term, then pick its group. Round 1 only reads off the letters. Round 2 is the real like-terms test.

Watch out: The number in front never decides it: 5a and 5x have the same number and are different kinds.

Predict, then check

Every bar below is the same length, and every bar is one a.

Here is 3a + 2a as bars: [a][a][a] + [a][a]. Before you read on: what does 3a + 2a come to?

Predict, then check

Picture algebra tiles. A green tile is +g and a red tile is −g. One green and one red cancel each other out: that's called a zero pair.

You have 1 green g tile and 5 red −g tiles on the table. Cancel every zero pair you can. What is g − 5g?

Maths · Simplifying

Four students, one expression

Four students are tidying up expressions. You can test any of their claims by trying numbers: here, f = 100 and g = 10.

Which idea is closest to what you think right now?

5f + 4g can be written as 9fg. You add the 5 and the 4, then stick the letters on.No, it isn't 9fg, it's 9fg². Adding makes the power go up.No. 5f + 4g is already as simple as it gets, because f and g are different kinds of term.4x + 3x ≡ 7x, so that tells me x = 7.

How sure are you?

I'm sureI think soI'm guessing

5f + 4g can be written as 9fg. You add the 5 and the 4, then stick the letters on.

This is a common misconception

Try it. With f = 100 and g = 10, 5f + 4g is 500 + 40 = 540. But 9fg is 9 × 100 × 10 = 9000. They are not the same, so they are not equivalent. f and g are different kinds of thing, like metres and kilograms, so there is nothing to collect. The 5 and the 4 only count how many of each kind. They can't be added to each other.

No, it isn't 9fg, it's 9fg². Adding makes the power go up.

This is a common misconception

Same test, same answer: with f = 100 and g = 10, 9fg² is 9 × 100 × 100 = 90 000, nowhere near 540. Adding never creates a power. When you add like terms you only count how many: 4x + 2x ≡ 6x, not 6x². The letter stays exactly as it was.

No. 5f + 4g is already as simple as it gets, because f and g are different kinds of term.

This matches the evidence

Exactly. Unlike terms stay side by side. 5f + 4g is already in its simplest form, and so are 2e + 9f and 5j + 8k.

4x + 3x ≡ 7x, so that tells me x = 7.

This is a common misconception

Simplifying and solving are different jobs. Simplifying only rewrites an expression in a neater form. 4x + 3x ≡ 7x is true whatever x is: try x = 1 (4 + 3 = 7) or x = 10 (40 + 30 = 70). Nothing there tells you what x is, and no equation has been solved.

Unlike terms stay side by side, adding like terms only changes how many (never the power), and simplifying rewrites an expression without solving it.

Worked example

Problem

Fully simplify c² + 8c − 3 + 3c² − 5c − 2

Maths · Your turn

Now you supply the steps

Job 1: A rectangle has sides 2x + 7y and 3x + 4y. Find its perimeter in its simplest form, then check it with x = 7 and y = 3. Job 2: four days' earnings at h pounds an hour, plus tips, come to 5h + 3h + 6h + 6h + 8 + 15 + 5 + 12 pounds. Is 8h + 20 + 12h + 20 fully simplified?

A rectangle has two pairs of equal sides, so its perimeter is all four sides added: (2x + 7y) + (3x + 4y) + (2x + 7y) + (3x + 4y).
missing step

Which line is step 2?

Collect the x terms: 2x + 3x + 2x + 3x = 10x.Collect the x terms: 2x + 3x + 2x + 3x = 10x².Collect the x terms: 2x + 3x = 5x.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

You can only add the same kind of thing.

What you need to know

Like terms have the same letters with the same powers. The number in front (the coefficient) can be anything, and it only counts how many.
You can only collect like terms. 3a + 2a ≡ 5a, but terms such as 5f and 4g are unlike, so they stay side by side.
To fully simplify, collect each family of like terms separately, then check that no family appears twice.
Simplifying rewrites an expression more neatly. It doesn't solve anything: 4x + 3x ≡ 7x does not mean x = 7.

The big picture

Like terms have the same letters with the same powers, and only the number in front can differ. You can collect like terms because they are the same kind of thing, but unlike terms stay side by side. To simplify fully, collect every family once.

Key points

1Same letters and same powers means like terms. The coefficient can be a fraction or negative, and a letter on its own, such as z, has an unwritten coefficient of 1.

2x, x² and x³ are three different kinds, and so are a²b and ab², because the powers have to match. Think of cm, cm² and cm³: different units.

3Collecting like terms is counting how many of one kind. Negative terms cancel positive ones: g − 5g ≡ −4g.

4Fully simplified means no like terms are left to collect. Rewriting the same terms in a different order doesn't count.

5A simplified expression is easier to use: put numbers in and the answer matches adding the original pieces.

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Worked example

Problem

Fully simplify 3a + 6b + 5a + b.

⚠ Watch out

Writing 4x + 2x as 6x². Adding only counts how many x's you have. It doesn't turn them into x × x, so the answer is 6x.

🧠

Memory hook

Think units. 4 cm and 8 cm are both centimetres, so 5a and 7a are both a's. But cm, cm² and cm³ are different units, just as x, x² and x³ are different terms. Same kind? Same letters, same powers. The number only counts.

✓

Check yourself

Which is a like term of 6x²: 6x, 2x², x³ or 6y²? Why? (Answer: 2x². Same letter, same power. Only the number differs.)

Flashcards

(12)

What makes two terms like terms?

The same letters with the same powers. The number in front can be different.

What is a coefficient?

The number in front of the letters, which counts how many. In −3e it is −3.

What is the coefficient of z?

1. We don't write it, because 1z is just z.

5a and 5x both have a 5. Why aren't they like terms?

The number in front doesn't decide it. The letters do, and a and x are different kinds.

Are x, x² and x³ like terms?

No. x² is x × x and x³ is x × x × x, so the powers differ, like cm, cm² and cm³.

Are a²b and ab² like terms?

No. In a²b the a is squared, and in ab² the b is squared, so the powers are not the same.

4de and 7bc both multiply two letters. Are they like terms?

No. The letters have to match, and d and e are not b and c.

What does the sign ≡ mean?

'Is identical to': true for every value of the letter. For example, 3a + 2a ≡ 5a.

Use zero pairs to find 5g − 4g.

Four green-and-red pairs cancel and one g is left: 5g − 4g ≡ g.

4x + 3x ≡ 7x. Does that mean x = 7?

No. Simplifying only rewrites the expression. It does not find a value for x.

What does 'fully simplified' mean?

No like terms are left to collect, so every family appears just once.

Why does simplifying help?

A short expression is quicker to use. Put numbers in and you get the same answer as adding the long list.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Same kind, or different kind?

Four students, one expression

Now you supply the steps

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