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Topic 2 of 5
AQAAlgebraPaper 1
Solving Linear Equations
AQAPaper 1Read
What is a linear equation?
A linear equation is a statement that two expressions are equal, where the unknown (usually x) appears only to the power of 1 — no x2, no square roots of x.
Definition
Solving an equation means finding the value of the unknown that makes both sides equal.
Golden rule
whatever you do to one side, do the same to the other
This keeps the equation balanced at every step.
One-step equations
A single inverse operation isolates x. Undo whatever is being done to x by applying the opposite operation to both sides.
+
x + 5 = 12 → subtract 5 from both sides → x = 7
−
x − 3 = 9 → add 3 to both sides → x = 12
×
4x = 20 → divide both sides by 4 → x = 5
÷
x3 = 6 → multiply both sides by 3 → x = 18
⭐ Examiner tip
Write out every step on its own line. Even if your final answer is wrong, you still pick up method marks for correct steps shown along the way.
Two-step equations
When two operations have been applied to x, undo them in reverse order — usually addition/subtraction first, then multiplication/division.
Worked example
Worked example
Solve 3x + 4 = 19
1
3x + 4 = 19
start
2
3x = 15
− 4 both sides
3
x = 5
÷ 3 both sides
Check: substitute x = 5 back in → 3(5) + 4 = 19 ✓
Quick recall · flip the card
To solve 3x + 4 = 19, which operation do you undo first?
Unknowns on both sides
When x appears on both sides of the equation, collect all the x terms onto one side and all the numbers onto the other, then solve as normal.
⚠ Common mistake
Students often forget to change the sign of a term when moving it across the equals sign. Moving +2x to the other side makes it −2x.
Equations with brackets
Expand the bracket first, then treat the equation as unknowns-on-both-sides.
Worked example
Worked example
Solve 5(x − 2) = 3x + 8
1
5x − 10 = 3x + 8
expand bracket
2
2x − 10 = 8
− 3x both sides
3
2x = 18
+ 10 both sides
4
x = 9
÷ 2 both sides
Check: substitute x = 9 back in → 5(9 − 2) = 35 and 3(9) + 8 = 35 ✓ both sides match
Equations with fractions
Multiply both sides by the denominator first to clear the fraction.
Worked example
Worked example
Solve x + 32 = 7
1
x + 3 = 14
× 2 both sides
2
x = 11
− 3 both sides
Check: (11 + 3) ÷ 2 = 7 ✓
Quick recall · flip the card
What must you do to both sides of an equation to keep it balanced?
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Expanding & Factorising
Key idea must-know fact or method|MCQ multiple choice|Fill in complete the blank|Match sort terms & definitions
