Two-Step Equations
Two-step equations form the foundation of algebraic problem-solving in Year 8, requiring students to perform inverse operations in the correct sequence. These equations, typically in the form ax + b = c, appear frequently in GCSE Foundation papers and real-world applications from calculating mobile phone bills to determining ticket prices.
Why it matters
Two-step equations connect abstract algebra to practical scenarios students encounter daily. When calculating the cost of school dinner vouchers (£2.50 per meal plus £3.50 delivery fee equals £13.50 total), students solve 2.5x + 3.5 = 13.5 to find x = 4 meals. Similarly, determining how many cinema tickets Amelia can buy with £45 when each ticket costs £8.75 plus a £1.25 booking fee requires solving 8.75x + 1.25 = 45, yielding x = 5 tickets. These skills prepare students for GCSE word problems involving perimeter calculations, age relationships, and consecutive number problems. The systematic approach of undoing operations in reverse order—addressing addition/subtraction before multiplication/division—develops logical reasoning essential for more complex algebraic manipulation in Key Stage 4.
How to solve two-step equations
Two-Step Equations
- Undo the addition/subtraction first (isolate the term with x).
- Then undo the multiplication/division.
- Verify by substituting back.
Example: 3x + 5 = 20 → 3x = 15 → x = 5.
Worked examples
Fill in the blank: 2 × ___ + 3 = 9
Answer: ___ = 3
- Rewrite as equation → 2x + 3 = 9 — The blank is our unknown x.
- Subtract 3 from both sides → 2 × ___ = 9 − 3 = 6 — Remove the constant.
- Divide both sides by 2 → ___ = 6 ÷ 2 = 3 — Find the missing value.
- Verify → 2 × 3 + 3 = 9 ✓ — Check the answer.
A student solved 6x + 6 = 12 like this: Step 1: 6x = 12 + 6 = 18 Step 2: x = 18 ÷ 6 = 3 Find and correct the error.
Answer: x = 1
- Identify the error → Step 1 is wrong: should subtract 6, not add it — To undo + 6, we subtract 6 from both sides.
- Correct Step 1 → 6x = 12 − 6 = 6 — Subtract the constant correctly.
- Correct Step 2 → x = 6 ÷ 6 = 1 — Divide to find x.
- Verify → 6·(1) + 6 = 6 + 6 = 12 ✓ — Substitute back to confirm.
Alfie is 2 times as old as Lily plus 2 years. Together they are 11 years old. How old is Lily?
Answer: Lily = 3
- Define variable → Let Lily's age = x, Alfie's age = 2x + 2 — Express Alfie's age in terms of Lily's.
- Write equation → x + (2x + 2) = 11 → 3x + 2 = 11 — Their ages sum to the total.
- Subtract 2 from both sides → 3x = 11 − 2 = 9 — Isolate the x term.
- Divide both sides by 3 → x = 9 ÷ 3 = 3 — Lily is 3 years old.
- Verify → Alfie = 2×3+2 = 8, 3+8 = 11 ✓ — Ages add up correctly.
Common mistakes
- Students often add the constant instead of subtracting when isolating the x term. For example, solving 4x + 7 = 23, they incorrectly write 4x = 23 + 7 = 30, giving x = 7.5 instead of the correct x = 4.
- Many pupils perform operations in the wrong order, dividing before removing the constant. When solving 3x + 6 = 21, they mistakenly calculate x + 2 = 7, then x = 5, rather than first finding 3x = 15, then x = 5.
- Students frequently forget to apply operations to both sides of the equation. Solving 5x - 8 = 12, they might write 5x = 12 + 8 on one line, then incorrectly state x = 12, omitting the division by 5.
- A common error involves incorrect verification, substituting the wrong value or making arithmetic mistakes. After finding x = 3 for 2x + 4 = 10, students might check 2(3) + 4 = 6 + 4 = 11, incorrectly concluding their answer is wrong.