Solving One-Step and Two-Step Equations
Use inverse operations to solve equations, keep both sides balanced, and check solutions by substitution.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
A cinema ticket booking costs $5 plus $12 per ticket. If the total is $41, how could you find the number of tickets? Without algebra — write your first strategy.
An equation is like a balance: both sides have the same value. To find the unknown, you undo the operations that were done to the variable — this is called using inverse operations.
One-step equations: $x + a = b$ — subtract $a$ from both sides. $ax = b$ — divide both sides by $a$.
Two-step equations: $ax + b = c$ — subtract $b$ first, then divide by $a$.
Key facts
- An equation states that two sides have equal value.
- Inverse operations undo each other.
- A solution is a value that makes the equation true.
Concepts
- Whatever is done to one side of an equation must be done to the other side.
- Two-step equations are solved by undoing addition or subtraction before multiplication or division.
- Substitution checks whether the solution is correct.
Skills
- Solve one-step equations such as $x + 7 = 19$ and $4x = 36$.
- Solve two-step equations such as $3x + 5 = 26$.
- Explain and check each solution.
An equation is like a balance: both sides have the same value. To keep the equation true, any operation used on one side must also be used on the other side. This is why each solving step changes both sides in the same way.
What to write in your book
- An equation is a balance: $\text{left side} = \text{right side}$.
- Inverse operations: addition ↔ subtraction; multiplication ↔ division.
- To isolate the variable: undo each operation done to it, applying the same operation to both sides.
- Always write the equation at the start of each solution, then show each balancing step.
Did you get this? True or false: to solve $x + 9 = 15$, you should add 9 to both sides.
For $3x + 5 = 26$, the variable is first multiplied by 3, then 5 is added. When solving, work backwards: undo the addition first, then undo the multiplication.
| Builds the expression | Solves the equation |
|---|---|
| $x \rightarrow 3x \rightarrow 3x + 5$ | $3x + 5 \rightarrow 3x \rightarrow x$ |
Key inverse operations reference:
| Equation type | Step to isolate variable |
|---|---|
| $x + a = b$ | subtract $a$ from both sides |
| $ax = b$ | divide both sides by $a$ |
| $ax + b = c$ | subtract $b$, then divide by $a$ |
What to write in your book
- Two-step solving order: undo addition/subtraction first, then undo multiplication/division.
- Think of building the expression forward ($x \to 3x \to 3x+5$) and solving backwards.
- After finding $x$, always substitute back to check: $3(7)+5 = 21+5 = 26$ ✓
Quick check: To solve $4x + 7 = 31$, what is the correct first step?
For an equation like $2(x + 4) = 18$, you can divide both sides by the coefficient of the bracket first, or expand the bracket first.
| Method A — divide first | Method B — expand first |
|---|---|
| $2(x + 4) = 18$ $x + 4 = 9$ $x = 5$ |
$2x + 8 = 18$ $2x = 10$ $x = 5$ |
Both methods give the same answer. Choose whichever is easier for the numbers in the question.
What to write in your book
- Brackets equation: either divide both sides by the coefficient first, or expand first.
- Check: $2(5 + 4) = 2(9) = 18$ ✓
- Write each step on a new line, aligned on the equals sign.
- Don't skip steps in an exam — each balancing operation can earn a method mark.
Fill the gap: To solve $3x + 5 = 26$: first subtract from both sides to get $3x =$ , then divide by 3 to get $x =$ .
Worked examples · 3 in a row, reveal as you go
Solve: (a) $x + 7 = 19$ (b) $4x = 36$
Solve $3x + 5 = 26$ and verify the solution by substitution.
Solve $2(x + 4) = 18$.
What to write in your book
- Always show each balancing step — write the operation applied (e.g. $-5$ on both sides).
- Always check by substituting your answer back in.
- For brackets: either divide by coefficient first, or expand first — both are correct.
Odd one out: Three of these are correct statements about solving equations. Which one is wrong?
Common errors · the 3 traps that cost marks
Quick-fire practice · 4 equations
Solve $y - 6 = 15$ and check your answer.
Solve $5a = 45$ and check your answer.
Solve $2m + 7 = 31$ and check your answer.
Solve $4(p - 3) = 28$ and check your answer.
Teach it back: In one or two sentences, explain why you undo addition before division when solving a two-step equation like $3x + 5 = 26$.
The cinema problem can be modelled by $5 + 12t = 41$, where $t$ is the number of tickets. Subtract 5 from both sides, then divide by 12: $t = 3$.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Solve $6x = 54$ and check your solution. (2 marks)
Q2. Solve $4x + 9 = 37$ and check your solution by substitution. (3 marks)
Q3. A taxi fare is $7 plus $3 per kilometre. The total fare is $31. Write and solve an equation for the number of kilometres. (4 marks)
📖 Comprehensive answers (click to reveal)
Drill 1: $y = 15 + 6 = 21$; check: $21 - 6 = 15$ ✓ · 2: $a = 45 \div 5 = 9$; check: $5 \times 9 = 45$ ✓ · 3: $2m = 24 \Rightarrow m = 12$; check: $2(12)+7=31$ ✓ · 4: $p - 3 = 7 \Rightarrow p = 10$; check: $4(10-3) = 28$ ✓
Q1 (2 marks): $x = 54 \div 6 = 9$ [1]; check: $6 \times 9 = 54$ ✓ [1].
Q2 (3 marks): $4x + 9 - 9 = 37 - 9$ [1]; $4x = 28 \Rightarrow x = 7$ [1]; check: $4(7) + 9 = 28 + 9 = 37$ ✓ [1].
Q3 (4 marks): Equation: $7 + 3k = 31$ (or $3k + 7 = 31$) [1]; $3k = 24$ [1]; $k = 8$ km [1]; check: $7 + 3(8) = 31$ ✓ [1].
Five timed questions on one-step and two-step equations. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering equation-solving questions. Pool: lesson 2.
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