Solving linear equations

A linear equation has only first-power x terms. Foundation, Intermediate and Higher all examine these.

Algebraic method — golden rule

Whatever you do to one side, do to the other. Aim to isolate x.

One-step equations

x + 7 = 12 → subtract 7: x = 5. 3x = 21 → divide by 3: x = 7.

Two-step equations

2x + 5 = 17.

• Subtract 5: 2x = 12.
• Divide by 2: x = 6.

Variables on both sides

5x − 4 = 2x + 11.

• Subtract 2x: 3x − 4 = 11.
• Add 4: 3x = 15.
• Divide by 3: x = 5.

Brackets

3(x + 4) = 21.

• Expand: 3x + 12 = 21. Or divide by 3 first: x + 4 = 7 → x = 3.

Fractional equations

(x + 3)/4 = 5.

• Multiply both sides by 4: x + 3 = 20 → x = 17.

For fractions on both sides, multiply through by the LCM of the denominators.

(x − 1)/2 = (x + 5)/3.

• LCM = 6. Multiply: 3(x − 1) = 2(x + 5).
• 3x − 3 = 2x + 10 → x = 13.

Graphical method

The solution to ax + b = 0 is the x-intercept of the line y = ax + b.

The solution to ax + b = cx + d is the x-coordinate of the intersection of y = ax + b and y = cx + d.

Verification

ALWAYS substitute back into the original equation as a check. WJEC awards no specific mark for verification but it catches errors that lose A1 marks.

WJEC exam tip

When both sides have variables, MOVE the smaller-coefficient term first to keep coefficients positive. 5x − 4 = 2x + 11 — subtract 2x (smaller coefficient) rather than 5x.