Solving Linear Equations with Brackets and Unknowns on Both Sides Revision Notes | GCSE AQA Higher Maths

1. Why Equation Solving Is the Engine of Algebra

Almost every algebra question on your GCSE paper — whether it involves geometry, sequences, or real-life problems — eventually asks you to solve an equation. Get this foundation right and a huge portion of the paper becomes manageable.

Think of an equation as a perfectly balanced set of scales. Whatever you do to one side, you must do to the other. That single rule is the entire logic of equation solving.

2. The Balance Method and Inverse Operations

Every operation has an inverse that undoes it. Addition is undone by subtraction; multiplication is undone by division. When you want to isolate $x$, you work backwards through whatever has been done to it, applying the inverse at each stage.

That one-step example shows the core idea. The moment you need two operations to reach $x$, you apply the inverses in reverse order — last operation undone first.

Worked example: Solving a Two-Step Equation

Solve $3x - 5 = 16$

1. $3x - 5 + 5 = 16 + 5$ $3x = 21$ — The last thing done to $x$ was subtracting 5, so we undo that first. [M1]
2. $\frac{3x}{3} = \frac{21}{3}$ $x = 7$ — Now undo the multiplication by 3. [A1]
3. $3(7) - 5 = 21 - 5 = 16$ ✓ — Substituting $x = 7$ into the original equation confirms equality. Always check In questions without a calculator.

Answer: $x = 7$

3. Equations with Brackets

When brackets appear in an equation, expand them first before applying the balance method. Expanding means multiplying the term outside the bracket by every term inside it.

Worked example: Solving an Equation with Brackets

Solve $5(2x - 3) = 25$

1. $5 \times 2x - 5 \times 3 = 25$ $10x - 15 = 25$ — Multiply 5 by both terms inside the bracket. [M1 for correct expansion]
2. $10x - 15 + 15 = 25 + 15$ $10x = 40$ — Undo the subtraction of 15.
3. $\frac{10x}{10} = \frac{40}{10}$ $x = 4$ — Undo the multiplication by 10. [A1]
4. $5(2 \times 4 - 3) = 5(8 - 3) = 5 \times 5 = 25$ ✓ — Confirms $x = 4$ is correct.

Answer: $x = 4$

4. Equations with Unknowns on Both Sides

Some equations have $x$ terms on both sides, such as $5x + 2 = 3x + 10$. The strategy is to collect all the $x$ terms onto one side and all the number terms onto the other.

Choose the side where the $x$ coefficient will be positive after collecting — this avoids dividing by a negative number later, which is a common source of sign errors.

Worked example: Solving an Equation with $x$ on Both Sides

Solve $5x + 2 = 3x + 10$

1. $5x + 2 - 3x = 3x + 10 - 3x$ $2x + 2 = 10$ — Collect $x$ terms on the left. Subtracting $3x$ keeps the coefficient positive. [M1]
2. $2x + 2 - 2 = 10 - 2$ $2x = 8$ — Move the number term to the right side.
3. $\frac{2x}{2} = \frac{8}{2}$ $x = 4$ — Isolate $x$. [A1]
4. Left: $5(4) + 2 = 20 + 2 = 22$ Right: $3(4) + 10 = 12 + 10 = 22$ ✓ — Both sides equal 22, confirming $x = 4$.

Answer: $x = 4$

5. Combining Skills: Brackets and Unknowns on Both Sides

Harder exam questions combine both skills — brackets appear alongside $x$ terms on both sides. The method is always the same: expand first, then collect, then solve.

Worked example: Brackets and Unknowns on Both Sides

Solve $4(x + 3) = 2x + 18$

1. $4x + 12 = 2x + 18$ — $4 \times x + 4 \times 3 = 4x + 12$. [M1 for correct expansion]
2. $4x + 12 - 2x = 2x + 18 - 2x$ $2x + 12 = 18$ — Collect $x$ terms on the left.
3. $2x + 12 - 12 = 18 - 12$ $2x = 6$ — Collect number terms on the right.
4. $\frac{2x}{2} = \frac{6}{2}$ $x = 3$ — Isolate $x$. [A1]
5. Left: $4(3 + 3) = 4 \times 6 = 24$ Right: $2(3) + 18 = 6 + 18 = 24$ ✓ — Both sides equal 24.

Answer: $x = 3$

6. Visual Model: The Balance in Action

Seeing the balance method visually reinforces why every operation must be applied to both sides. The diagram below represents $2x + 3 = 11$ as a set of scales — subtracting 3 from both sides tips neither side, keeping the equation true.

7. Exam Technique and Common Mistakes

The methods above are straightforward when applied carefully, but certain patterns trip students up repeatedly in exams. A common mistake is to undo operations in the wrong order: for example, in $5(2x-3)=25$, you must expand or divide before you try to add or subtract. Another mistake is forgetting that a negative sign applies to every term inside brackets. Students also lose marks by stopping as soon as they find $x$ without checking the original equation. A quick substitution check is often the fastest way to catch arithmetic slips before they cost marks. Building the habit of checking each step keeps equation solving accurate and efficient.

Mastering one-step equations, two-step equations, brackets, and unknowns on both sides gives you the algebraic foundation needed for later GCSE topics. These methods reappear in ratio, geometry, graphs, and formula work, so fluency here matters. The key idea is always the same: keep the equation balanced, reverse operations carefully, and write a clear final answer that can be checked in the original statement.