Solving Linear Equations
The Skill
A linear equation contains a variable (usually $x$) raised to the power of 1. Solving means finding the value of the variable that makes the equation true.
The Golden Rule
Whatever you do to one side, you must do to the other side.
One-Step Equations
$$x + 5 = 12$$ Subtract 5 from both sides: $$x = 7$$
$$3x = 15$$ Divide both sides by 3: $$x = 5$$
Two-Step Equations
$$2x + 3 = 11$$
Step 1: Subtract 3 from both sides $$2x = 8$$
Step 2: Divide both sides by 2 $$x = 4$$
Equations with Unknowns on Both Sides
$$5x + 2 = 3x + 10$$
Step 1: Get all $x$ terms on one side (subtract $3x$ from both sides) $$2x + 2 = 10$$
Step 2: Subtract 2 from both sides $$2x = 8$$
Step 3: Divide by 2 $$x = 4$$
Equations with Brackets
$$3(x + 4) = 21$$
Method 1: Expand first $$3x + 12 = 21$$ $$3x = 9$$ $$x = 3$$
Method 2: Divide first (if possible) $$x + 4 = 7$$ $$x = 3$$
The Traps
Common misconceptions and how to avoid them.
Errors when collecting x terms "The Collection Confusion"
The Mistake in Action
Solve $5x + 2 = 2x + 14$
Wrong: $5x + 2x = 14 + 2$ $7x = 16$
Why It Happens
Students try to get all x terms on one side but add instead of subtract (or vice versa).
The Fix
Be systematic: choose one side for x terms and one side for number terms.
Correct method: $$5x + 2 = 2x + 14$$
Subtract $2x$ from both sides (collecting x on the left): $$3x + 2 = 14$$
Subtract 2 from both sides: $$3x = 12$$
Divide by 3: $$x = 4$$
Check: LHS = $5(4) + 2 = 22$, RHS = $2(4) + 14 = 22$ ✓
Spot the Mistake
Solve $5x + 2 = 2x + 14$
$5x + 2x = 14 + 2$
$7x = 16$
Click on the line that contains the error.
Forgetting to change the sign when moving terms "The Sign Slip"
The Mistake in Action
Solve $x + 7 = 12$
Wrong: $x = 12 + 7$ $x = 19$
Why It Happens
Students "move" the 7 to the other side but forget that moving involves the inverse operation. They keep the + sign instead of changing it to −.
The Fix
Don't think of "moving" terms — think of doing the inverse operation to both sides.
To remove "+7" from the left side, subtract 7 from both sides: $$x + 7 = 12$$ $$x + 7 - 7 = 12 - 7$$ $$x = 5$$
Memory aid: When a term "crosses the bridge" (equals sign), it changes its sign.
Spot the Mistake
Solve $x + 7 = 12$
$x = 12 + 7$
$x = 19$
Click on the line that contains the error.
Only doing the operation on one side "The One-Sided Operation"
The Mistake in Action
Solve $3x = 18$
Wrong: $3x \div 3 = 18$ $x = 18$
Why It Happens
Students correctly identify that they need to divide by 3, but only apply it to the left side.
The Fix
Whatever you do to one side, you MUST do to the other side. The equation is like a balance scale — both sides must be treated equally.
$$3x = 18$$ $$\frac{3x}{3} = \frac{18}{3}$$ $$x = 6$$
Check: $3 \times 6 = 18$ ✓
Spot the Mistake
Solve $3x = 18$
$3x \div 3 = 18$
$x = 18$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Solve $4x - 7 = 17$
Solution
Step 1: Add 7 to both sides (to isolate the x term) $$4x - 7 + 7 = 17 + 7$$ $$4x = 24$$
Step 2: Divide both sides by 4 $$\frac{4x}{4} = \frac{24}{4}$$ $$x = 6$$
Check: $4(6) - 7 = 24 - 7 = 17$ ✓
Answer: $x = 6$
Question
Solve $7x - 5 = 3x + 11$
Solution
Step 1: Collect x terms on the left by subtracting $3x$ from both sides $$7x - 3x - 5 = 11$$ $$4x - 5 = 11$$
Step 2: Add 5 to both sides $$4x = 16$$
Step 3: Divide both sides by 4 $$x = 4$$
Check:
- LHS: $7(4) - 5 = 28 - 5 = 23$
- RHS: $3(4) + 11 = 12 + 11 = 23$ ✓
Answer: $x = 4$
Question
Solve $5(2x - 3) = 25$
Solution
Method 1: Expand first $$10x - 15 = 25$$ $$10x = 40$$ $$x = 4$$
Method 2: Divide first (since 25 is divisible by 5) $$2x - 3 = 5$$ $$2x = 8$$ $$x = 4$$
Check: $5(2(4) - 3) = 5(8 - 3) = 5(5) = 25$ ✓
Answer: $x = 4$
Level 2: Scaffolded
Fill in the key steps.
Question
Solve $\frac{x - 4}{3} = 7$
Level 3: Solo
Try it yourself!
Question
Solve $4(x + 3) = 2(x + 9)$
Show Solution
Expand both brackets: $$4x + 12 = 2x + 18$$
Subtract $2x$ from both sides: $$2x + 12 = 18$$
Subtract 12 from both sides: $$2x = 6$$
Divide by 2: $$x = 3$$
Check:
- LHS: $4(3 + 3) = 4(6) = 24$
- RHS: $2(3 + 9) = 2(12) = 24$ ✓
Answer: $x = 3$
Question
The perimeter of a rectangle is 46 cm. The length is $x$ cm and the width is $(x - 5)$ cm. Find the value of $x$.
Show Solution
Perimeter = $2 \times \text{length} + 2 \times \text{width}$
$$2x + 2(x - 5) = 46$$ $$2x + 2x - 10 = 46$$ $$4x - 10 = 46$$ $$4x = 56$$ $$x = 14$$
Check: Length = 14cm, Width = 9cm, Perimeter = 2(14) + 2(9) = 28 + 18 = 46cm ✓
Answer: $x = 14$
Examiner's View
Mark allocation: Linear equations are typically 2-4 marks depending on complexity.
Common errors examiners see:
- Sign errors when moving terms across the equals sign
- Forgetting to apply operations to both sides
- Errors when expanding brackets
- Not showing working (losing method marks)
What gains marks:
- Show each step clearly on a new line
- Write "= ..." to start each new line
- Check your answer by substituting back
AQA Notes
AQA likes to include equations in context (e.g., "The perimeter is 34cm, find x").