Two-Step Equations
A two-step equation contains one variable that requires exactly two operations to isolate. These equations follow the form ax + b = c or ax - b = c, where a represents the coefficient of the variable, b is a constant, and c is the result. The solution process involves systematically undoing operations in reverse order.
Why it matters
Two-step equations model countless real-world scenarios across finance, science, and daily life. A mobile phone contract might charge £25 monthly plus a £50 connection fee — after 6 months, the total cost equation becomes 25x + 50 = 200. Construction workers use similar equations when calculating materials: if concrete costs £80 per cubic metre plus a £120 delivery fee, and the total bill is £440, then 80x + 120 = 440 determines the volume ordered. These equations appear throughout GCSE mathematics, forming the foundation for more complex algebraic concepts including simultaneous equations, quadratic functions, and linear programming. Mastering two-step equations in Year 8 prepares students for advanced topics where multiple variables and higher-degree polynomials require similar systematic approaches to isolation and solution.
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
I think of a number, multiply it by 2 and add 2. I get 4. What is my number?
Answer: x = 1
- Write as an equation → 2x + 2 = 4 — Let x be the number. 'Multiply by a and add b gives c' becomes ax + b = c.
- Subtract 2 from both sides → 2x = 4 − 2 = 2 — Isolate the term with x.
- Divide both sides by 2 → x = 2 ÷ 2 = 1 — Find x.
- Verify → 2 × 1 + 2 = 2 + 2 = 4 ✓ — Check: the number works.
A phone plan costs £3.00 per month plus a £2.00 signup fee. After one month the total cost is £23.00. Write and solve the equation to confirm the monthly cost.
Answer: x = 7
- Write the equation → 3x + 2 = 23, where x = number of months — Monthly cost times months plus signup fee equals total.
- Subtract 2 from both sides → 3x = 23 − 2 = 21 — Remove the signup fee.
- Divide both sides by 3 → x = 21 ÷ 3 = 7 — x = 7 month(s), confirming £3.00/month.
Solve for x: 6x − 5 = -41
Answer: x = -6
- Identify the goal → 6x − 5 = -41 — solve for x — We want to get x alone on one side. This takes two steps: first remove the constant, then remove the coefficient.
- Step 1: Add 5 to both sides → 6x − 5 + 5 = -41 + 5 → 6x = -36 — Undo the addition/subtraction to isolate the term with x.
- Step 2: Divide both sides by 6 → 6x ÷ 6 = -36 ÷ 6 → x = -6 — Undo the multiplication. -36 ÷ 6 = -6.
- Verify by substituting back → 6·(-6) − 5 = -36 − 5 = -41 ✓ — Replace x with our answer in the original equation. Both sides should be equal.
Common mistakes
- Performing operations in the wrong order — attempting 3x + 7 = 22 by dividing first gives (3x + 7) ÷ 3 = 22 ÷ 3, yielding x + 7/3 = 22/3 instead of the correct x = 5.
- Applying operations to only one side of the equation — solving 4x - 9 = 15 by adding 9 to the right side only gives 4x - 9 = 24, not the correct 4x = 24.
- Sign errors when dealing with subtraction — solving 2x - 8 = -14 incorrectly as 2x = -14 - 8 = -22, giving x = -11 instead of the correct x = -3.