Solving Linear Equations

The Golden Rule: The Balance Method

Think of an equation like a set of weighing scales. The equals sign \(=\) means both sides are perfectly balanced.

Whatever operation you do to one side, you MUST do to the other side.

Your goal is to isolate \(x\) (get \(x\) on its own) by performing the opposite operations in reverse order.

1. Two-Step Equations

Usually, you have a multiplication and an addition/subtraction to deal with. Undo the addition/subtraction first.

Solve \( 3x + 5 = 17 \) Step 1: Remove the +5 (Subtract 5 from both sides) \( 3x = 17 - 5 \)
\( 3x = 12 \) Step 2: Remove the 3 (Divide both sides by 3) \( x = 12 \div 3 \) \( x = 4 \)

2. Unknowns on Both Sides

If \(x\) appears on the left and the right, you need to eliminate the smaller \(x\) term first.

Solve \( 5x - 2 = 3x + 8 \) Step 1: Eliminate the smallest x term (\(3x\)) Subtract \(3x\) from both sides:
\( 2x - 2 = 8 \) Step 2: Solve like a normal two-step equation Add 2 to both sides:
\( 2x = 10 \)
Divide by 2:
\( x = 5 \)

3. Brackets and Fractions

If the equation looks messy, tidy it up first!

Brackets: Expand them (multiply them out) first.
Fractions: Multiply to get rid of the denominator.

Solve \( \frac{x + 4}{3} = 5 \) Step 1: Multiply by 3 to remove the fraction \( x + 4 = 15 \) Step 2: Subtract 4 \( x = 11 \)

Practice Questions & Solutions

Try these yourself, then click to check your working.

Q1: \( x + 9 = 20 \)

Subtract 9 from both sides.
\( x = 20 - 9 \)
\( x = 11 \)

Q2: \( 4x = 32 \)

Divide both sides by 4.
\( x = 32 \div 4 \)
\( x = 8 \)

Q3: \( 2x - 7 = 13 \)

1. Add 7 to both sides.
\( 2x = 20 \)
2. Divide by 2.
\( x = 10 \)

Q4: \( \frac{x}{5} + 3 = 9 \)

1. Subtract 3 first.
\( \frac{x}{5} = 6 \)
2. Multiply by 5.
\( x = 6 \times 5 \)
\( x = 30 \)

Q5: \( 7x + 4 = 5x + 16 \)

1. Subtract \(5x\) from both sides (remove smaller \(x\)).
\( 2x + 4 = 16 \)
2. Subtract 4.
\( 2x = 12 \)
3. Divide by 2.
\( x = 6 \)

Q6: \( 3(x + 5) = 21 \)

1. Expand the bracket: \( 3 \times x \) and \( 3 \times 5 \).
\( 3x + 15 = 21 \)
2. Subtract 15.
\( 3x = 6 \)
3. Divide by 3.
\( x = 2 \)

Q7: \( 6x - 10 = 2x + 14 \)

1. Subtract \(2x\) from both sides.
\( 4x - 10 = 14 \)
2. Add 10 to both sides.
\( 4x = 24 \)
3. Divide by 4.
\( x = 6 \)

Q8: \( \frac{3x - 2}{4} = 4 \)

1. Multiply by 4 (remove the fraction).
\( 3x - 2 = 16 \)
2. Add 2.
\( 3x = 18 \)
3. Divide by 3.
\( x = 6 \)

Q9: \( 2(3x - 1) = 4(x + 2) \)

1. Expand both brackets.
\( 6x - 2 = 4x + 8 \)
2. Subtract \(4x\) (smallest \(x\)).
\( 2x - 2 = 8 \)
3. Add 2.
\( 2x = 10 \)
4. Divide by 2.
\( x = 5 \)

Q10: Trick Question - \( \frac{10}{x} = 2 \)

\(x\) is on the bottom! We hate that.
1. Multiply both sides by \(x\).
\( 10 = 2x \)
2. Now it's easy. Divide by 2.
\( 5 = x \) (or \( x = 5 \))