Two-Step Equations

A two-step equation needs exactly two operations to isolate the variable. Undo addition or subtraction first, then undo multiplication or division.

What is a Two-Step Equation?

A two-step equation is an equation you can solve using exactly two inverse operations.

$3x + 5 = 20$

Two things are happening to $x$ : it is multiplied by 3 then 5 is added. To solve, undo those operations in reverse order.

The Two Steps

Step 1 — Undo addition or subtraction (get the term with $x$ alone on one side). Step 2 — Undo multiplication or division (isolate $x$ ).

Solve 3x + 5 = 20

Step 1: Subtract 5 from both sides: $3x = 15$

Step 2: Divide both sides by 3: $x = 5$

Check: $3(5) + 5 = 15 + 5 = 20$ ✓

Solve 4x − 7 = 13

Step 1: Add 7 to both sides: $4x = 20$

Step 2: Divide both sides by 4: $x = 5$

Check: $4(5) − 7 = 20 − 7 = 13$ ✓

Equations with Fractions

When $x$ is divided, multiplication is used in step 2.

Solve x/3 + 2 = 8

Step 1: Subtract 2: $x/3 = 6$

Step 2: Multiply by 3: $x = 18$

Check: $18 ÷ 3 + 2 = 6 + 2 = 8$ ✓

Why Check Your Answer?

Always substitute your answer back into the original equation. If both sides are equal, the solution is correct. This catches small arithmetic errors.

Real-World Applications

Real world: Budgeting pocket money

You have €50 in savings. After buying a book for €8, you spend the rest equally on 6 cinema tickets. How much is each ticket? $6t + 8 = 50 \implies 6t = 42 \implies t = 7\text{ EUR}$ per ticket.

Real world: Overnight temperature

By midnight the temperature had risen 4°C from the evening reading and reached 9°C. What was the evening temperature? $x + 4 = 9 \implies x = 5°C$ — straightforward two-step reasoning used in weather forecasts.

Real world: Splitting a restaurant bill

Four friends split a meal equally and each pays a €1.50 booking fee on top. Each person pays €13 total. Two steps: $\frac{b}{4} + 1.50 = 13 \implies \frac{b}{4} = 11.50 \implies b = 46\text{ EUR}$ was the full bill.

Tip

Where do you see this? Splitting a bill with a service charge, calculating how many items you can afford from a budget, reading temperature changes on a weather app.

Practice Problems

Solve: 2x + 9 = 17

Show answer

Step 1: 2x = 8. Step 2: x = 4. Check: 2(4)+9 = 17 ✓

Solve: 5x − 3 = 22

Show answer

Step 1: 5x = 25. Step 2: x = 5. Check: 5(5)−3 = 22 ✓

Solve: x/4 + 6 = 10

Show answer

Step 1: x/4 = 4. Step 2: x = 16. Check: 16/4 + 6 = 4 + 6 = 10 ✓