To solve a two-step equation, identify the two operations applied to the unknown, then undo them in reverse order using inverse operations — keeping the equation balanced by doing the same to both sides. For example, to solve 2x + 3 = 11, first subtract 3 from both sides, then divide both sides by 2, giving x = 4.
What is a two-step equation?
A two-step equation is one where you need exactly two inverse operations to isolate the unknown. Common forms include:
ax + b = c(multiply then add)x/a + b = c(divide then add)a(x + b) = c(multiply a bracket)
The goal is always to get the variable (usually x) on its own on one side of the equals sign.
What are inverse operations?
Inverse operations are pairs that undo each other:
| Operation | Inverse |
|---|---|
| Addition (+) | Subtraction (−) |
| Subtraction (−) | Addition (+) |
| Multiplication (×) | Division (÷) |
| Division (÷) | Multiplication (×) |
When solving, always undo operations in reverse order — the last thing done to x is the first to be undone.
How to solve a two-step equation step by step
- Identify the two operations applied to the unknown.
- Undo the addition or subtraction first (the outermost operation).
- Undo the multiplication or division second.
- Check by substituting your answer back into the original equation.
Worked example 1 — 2x + 3 = 11
The equation applies multiply by 2, then add 3 to x.
- Step 1: Subtract 3 from both sides:
2x + 3 − 3 = 11 − 3, giving2x = 8 - Step 2: Divide both sides by 2:
2x ÷ 2 = 8 ÷ 2, givingx = 4
Check: 2(4) + 3 = 8 + 3 = 11 ✓
Worked example 2 — 5x − 7 = 23
- Step 1: Add 7 to both sides:
5x − 7 + 7 = 23 + 7, giving5x = 30 - Step 2: Divide both sides by 5:
5x ÷ 5 = 30 ÷ 5, givingx = 6
Check: 5(6) − 7 = 30 − 7 = 23 ✓
Worked example 3 — x/3 + 4 = 10
- Step 1: Subtract 4 from both sides:
x/3 = 6 - Step 2: Multiply both sides by 3:
x = 18
Check: 18/3 + 4 = 6 + 4 = 10 ✓
How do you solve equations with negative numbers?
The same balance method applies — just be careful with signs.
Worked example 4 — 3x − 8 = −2
- Step 1: Add 8 to both sides:
3x = −2 + 8 = 6 - Step 2: Divide both sides by 3:
x = 2
Check: 3(2) − 8 = 6 − 8 = −2 ✓
Worked example 5 — −4x + 1 = 13
- Step 1: Subtract 1 from both sides:
−4x = 12 - Step 2: Divide both sides by −4:
x = 12 ÷ (−4) = −3
Check: −4(−3) + 1 = 12 + 1 = 13 ✓
Remember: dividing by a negative number flips the sign of the result.
How do you solve equations with brackets?
Equations like 4(x + 2) = 20 can be solved in two ways: expand the bracket first, or divide both sides first.
Method 1 — expand first:
4(x + 2) = 20
Expand: 4x + 8 = 20
Subtract 8: 4x = 12
Divide by 4: x = 3
Method 2 — divide first (often quicker):
4(x + 2) = 20
Divide both sides by 4: x + 2 = 5
Subtract 2: x = 3
Both give x = 3. Choose whichever method keeps the numbers simpler.
Solving equations with fractions on one side
If the unknown is in the numerator of a fraction, multiply first.
Worked example 6 — (x − 1)/5 = 3
- Multiply both sides by 5:
x − 1 = 15 - Add 1 to both sides:
x = 16
Check: (16 − 1)/5 = 15/5 = 3 ✓
Common mistakes to avoid
| Mistake | Error | Correction |
|---|---|---|
| Wrong order of inverse operations | Dividing before subtracting | Undo add/subtract before multiply/divide |
| Forgetting to apply operations to both sides | 2x + 3 = 11 → 2x = 11 (missing −3) |
Always do the same to both sides |
| Sign error with negative coefficient | −4x = 12 → x = 12 |
Divide by −4: x = −3 |
| Not checking the answer | Leaving without substitution check | Always substitute back in |
Why does this matter for GCSE?
Two-step equations are the foundation for harder GCSE algebra, including equations with unknowns on both sides (3x + 2 = x + 10), equations formed from geometry or probability, and rearranging formulae. The DfE's KS3 maths programme of study lists solving linear equations as a core requirement, and exam boards build on this directly in Years 10 and 11. Fluency at the two-step level — especially the habit of checking — means fewer errors under pressure.
Frequently asked questions
How do you solve a two-step equation?
Undo the two operations applied to the unknown in reverse order, doing the same to both sides each time. First undo any addition or subtraction, then undo any multiplication or division. Always check by substituting your answer back into the original equation.
What does the balance method mean?
The balance method means treating the equals sign like the centre of a set of scales — whatever you do to one side, you must do exactly the same to the other. This keeps the equation true and eventually isolates the unknown.
What is an inverse operation?
An inverse operation is one that undoes another. Addition and subtraction are inverses; multiplication and division are inverses. When solving equations, you use the inverse of each operation to reverse what was done to the variable.
How do you check the answer to an equation?
Substitute your value for x back into the original equation and work out both sides. If both sides give the same number, your answer is correct. If not, you have made an error somewhere in the working.
What happens if the coefficient of x is negative?
You still divide both sides by the coefficient, but dividing by a negative number changes the sign of the result. For example, if −3x = 12, then x = 12 ÷ (−3) = −4.
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