Algebra · GCSE Maths

Algebra

Collect like terms, expand and factorise, and solve linear equations with the balance method: whatever you do to one side, do to the other.

UNDERSTAND→RETRIEVE→REMEMBER
THE MEMORY HOOK
The equals sign is a balance, not a button that means “the answer is”. Whatever you do to one side, do to the other.

The important bits

What you need to know

  1. 1

    Like terms have the same letter and the same power. 3x and 5x can be collected; 3x and 3x² cannot.

  2. 2

    Expanding means multiplying every term inside a bracket by the term outside. 3(2x − 5) = 6x − 15. With two brackets, use FOIL or a grid.

  3. 3

    Factorising is expanding in reverse. Highest common factor first: 6x + 9 = 3(2x + 3). Quadratics need two brackets when they factorise over the integers.

  4. 4

    Solve linear equations with the balance method. Subtract, add, divide or multiply on both sides until x is alone.

  5. 5

    Undo operations in reverse order. If x was multiplied by 3 and then 4 was added, subtract 4 first, then divide by 3.

  6. 6

    Equations with x on both sides: collect x terms on one side and numbers on the other. 5x − 2 = 3x + 10 becomes 2x = 12.

  7. 7

    Changing the subject uses the same balance. To make c the subject of 2(c + 5) = d, divide by 2, then subtract 5.

  8. 8

    Form an equation from a worded question before you solve. Define the unknown in a sentence: “Let x be the smaller number.”

Go deeper

Balance, do not teleport

The equals sign states that two expressions have the same value. If you add 7 to the left-hand side and forget the right, the statement becomes false. That is why the balance method is not a classroom slogan; it is the definition of equation. Inverse operations undo each other: addition undoes subtraction, multiplication undoes division, and a square root undoes a square (with a sign warning). To solve 3x + 5 = 17, subtract 5 from both sides to get 3x = 12, then divide both sides by 3 to get x = 4. Write the new equation after every step. An examiner who can follow 3x = 12 can still award method if the last division is wrong. A jump from 3x + 5 = 17 to x = 4 with no middle line cannot.

Go deeper

Expanding and factorising are the same door

Expanding 4(x − 3) produces 4x − 12 because both terms inside the bracket are multiplied by 4. The minus belongs to the 3; it is not a decoration on the bracket. Factorising 4x − 12 asks what was multiplied: 4 is common, so 4(x − 3). For two brackets, (x + 2)(x + 5) = x² + 7x + 10. The 7 comes from adding 2 and 5; the 10 comes from multiplying them. Factorising x² + 7x + 10 is the reverse hunt: two numbers that multiply to 10 and add to 7. If the quadratic is x² − 5x + 6, the numbers are −2 and −3, because negatives still multiply to a positive. Always expand your factorisation immediately as a check. If you do not recover the original quadratic, the brackets are wrong, however neat they look.

Go deeper

Unknowns on both sides still balance

When x appears twice, the aim is still a true equation with x isolated. For 5x − 2 = 3x + 10, subtract 3x from both sides: 2x − 2 = 10. Add 2 to both sides: 2x = 12. Divide by 2: x = 6. Check in the original: left side 5(6) − 2 = 28; right side 3(6) + 10 = 28. The check is not optional on a paper that offers an answer line worth one mark after three method marks. Inequalities use the same moves, with one extra rule: multiplying or dividing by a negative reverses the inequality sign. Fractions in equations are cleared by multiplying through by the denominator, still on both sides. The method does not change because the algebra looks heavier; the balance does not care about fractions.

WORKED EXAMPLE

See the idea in action

Solve 4(x − 3) = 2x + 6. Expand: 4x − 12 = 2x + 6. Subtract 2x from both sides: 2x − 12 = 6. Add 12 to both sides: 2x = 18. Divide by 2: x = 9. Check: left 4(9 − 3) = 24; right 2(9) + 6 = 24.

Exam technique

Turn knowledge into marks

After solving, substitute back into the original equation, not into a later line you might have copied wrongly. The check costs ten seconds and protects four marks.

Common mistakes

Do not give these marks away

  1. 01

    Subtracting a term from only one side, then wondering why the check fails.

  2. 02

    Writing 3(x + 4) as 3x + 4, forgetting to multiply the second term.

  3. 03

    Moving a term across the equals sign and changing its sign, then doing the inverse operation as well — a double move.

QUICK RETRIEVAL

What is the solution of 5x − 2 = 3x + 10?

Ax = 4

Bx = 6

Cx = 12

Dx = 2

Show the answer

x = 6. Subtract 3x from both sides: 2x − 2 = 10. Add 2: 2x = 12. Divide by 2: x = 6. Check: 5(6) − 2 = 28 and 3(6) + 10 = 28.

Quick questions

If this is the bit you searched

Why do teachers insist on writing a new line for each step?

Because method marks sit on those lines. The balance method is also easier to check: each line must remain a true equation. Jumps hide sign errors.

How do I change the subject of a formula?

Use the same inverse operations as solving. If A = πr², divide both sides by π, then square-root both sides to make r the subject. Write r = √(A/π).

What does “like terms” actually mean?

Terms you can add because they count the same kind of thing. 7xy and −2xy are like; 7xy and 7x are not, because one has y and the other does not.