Equations

The balance method

An equation stays true if you do the same thing to both sides. Undo operations in reverse order until the pronumeral is alone:

$$10 + 3x = 28 \implies 3x = 18 \implies x = 6$$

Write every step on a new line — these exams allocate a mark for working and a mark for the solution, so a bare answer earns half.

Equations with fractions

Multiply both sides by the denominator to clear the fraction, and multiply every term:

$$\frac{2x}{4} - 1 = 2 \implies 2x - 4 = 8 \implies 2x = 12 \implies x = 6$$

With two fractions, multiply through by a common denominator of both. If the whole side is one fraction, such as $\frac{x-4}{7} = 6$, one multiplication clears it: $x - 4 = 42$, so $x = 46$.

Brackets on both sides

Expand all brackets first, then gather the pronumeral terms on one side and numbers on the other:

$$7(3 - x) = 5(x - 3) \implies 21 - 7x = 5x - 15 \implies 36 = 12x \implies x = 3$$

Watch the signs when a negative multiplies through a bracket.

Check by substitution

Substitute your answer into both sides of the original equation — not a later line — and confirm they match. This same skill answers “spot the mistake” questions: test each line of the given working until two neighbouring lines disagree, then name the line and redo it correctly.

Word problems

Turn the story into an equation before solving:

1. Define the pronumeral: “let the smallest number be $n$”.
2. Consecutive numbers are $n$, $n+1$, $n+2$; consecutive even (or odd) numbers are $n$, $n+2$, $n+4$.
3. Write the equation, solve it, then answer the actual question asked.

Three consecutive even numbers summing to 48: $n + (n+2) + (n+4) = 48$, so $3n + 6 = 48$, $n = 14$ — the numbers are 14, 16 and 18.