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§ Algebra

Two-Step Equations

CCSS.7.EECCSS.8.EE4 min read

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.

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§ 01

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.

§ 02

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.

§ 03

Worked examples

Beginner§ 01

Fill in the blank: 2 × ___ + 3 = 9

Answer: ___ = 3

  1. Rewrite as equation → 2x + 3 = 9 — The blank is our unknown x.
  2. Subtract 3 from both sides → 2 × ___ = 9 − 3 = 6 — Remove the constant.
  3. Divide both sides by 2 → ___ = 6 ÷ 2 = 3 — Find the missing value.
  4. Verify → 2 × 3 + 3 = 9 ✓ — Check the answer.
Easy§ 02

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

  1. Identify the error → Step 1 is wrong: should subtract 6, not add it — To undo + 6, we subtract 6 from both sides.
  2. Correct Step 1 → 6x = 12 − 6 = 6 — Subtract the constant correctly.
  3. Correct Step 2 → x = 6 ÷ 6 = 1 — Divide to find x.
  4. Verify → 6·(1) + 6 = 6 + 6 = 12 ✓ — Substitute back to confirm.
Medium§ 03

Alfie is 2 times as old as Lily plus 2 years. Together they are 11 years old. How old is Lily?

Answer: Lily = 3

  1. Define variable → Let Lily's age = x, Alfie's age = 2x + 2 — Express Alfie's age in terms of Lily's.
  2. Write equation → x + (2x + 2) = 11 → 3x + 2 = 11 — Their ages sum to the total.
  3. Subtract 2 from both sides → 3x = 11 − 2 = 9 — Isolate the x term.
  4. Divide both sides by 3 → x = 9 ÷ 3 = 3 — Lily is 3 years old.
  5. Verify → Alfie = 2×3+2 = 8, 3+8 = 11 ✓ — Ages add up correctly.
§ 04

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.
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§ 05

Frequently asked questions

Why do we subtract the constant before dividing by the coefficient?
We follow the reverse order of operations (BIDMAS backwards). Since the expression 3x + 7 means 'multiply by 3, then add 7', we undo this by subtracting 7 first, then dividing by 3. This systematic approach ensures we isolate x correctly every time.
How do I handle negative coefficients in two-step equations?
Treat negative coefficients like positive ones but remember the sign rules. For -4x + 9 = 1, subtract 9 from both sides to get -4x = -8, then divide by -4 to find x = 2. Always check your answer by substituting back into the original equation.
What if my answer is a fraction or decimal?
Fractional and decimal answers are perfectly valid. When solving 3x + 5 = 7, you get 3x = 2, so x = 2/3. Always verify by substituting: 3(2/3) + 5 = 2 + 5 = 7 ✓. Convert between fractions and decimals as needed for verification.
How do I set up word problems as two-step equations?
Identify the unknown quantity (let x = ...), then translate the words into mathematical operations. 'Three times a number plus 8 equals 23' becomes 3x + 8 = 23. Look for key phrases like 'plus', 'increased by', 'times', and 'equals' to guide your equation setup.
What's the best way to check if my solution is correct?
Substitute your answer back into the original equation and verify both sides are equal. If solving 2x - 3 = 11 gives x = 7, check: 2(7) - 3 = 14 - 3 = 11 ✓. If the equation doesn't balance, revisit your working for arithmetic errors or incorrect operations.
§ 06

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