Solving linear equations

Notes

Key idea: an equation is a balance. Whatever you do to one side, you must do to the other. Solving means getting the unknown on its own: $x = \dots$

The balance method

Undo operations in the reverse order from the order they were applied. Think of the unknown being "wrapped up" and you are unwrapping it.

Example 1: solve $3x + 5 = 20$.

$$3x + 5 = 20$$ $$3x = 15 \quad (\text{subtract } 5)$$ $$x = 5 \quad (\text{divide by } 3)$$

Check by substituting back: $3 \times 5 + 5 = 20$. Correct.

Unknowns on both sides

Collect the $x$ terms on one side (choose the side so the $x$ coefficient stays positive), and the numbers on the other.

Example 2: solve $7x - 4 = 3x + 12$.

$$7x - 4 = 3x + 12$$ $$4x - 4 = 12 \quad (\text{subtract } 3x)$$ $$4x = 16 \quad (\text{add } 4)$$ $$x = 4 \quad (\text{divide by } 4)$$

Brackets

Expand brackets first, then solve as usual.

Example 3: solve $3(2x - 1) = 21$.

$$6x - 3 = 21$$ $$6x = 24$$ $$x = 4$$

Fractions

Multiply both sides by the denominator to remove the fraction.

Example 4: solve $\frac{x + 3}{2} = 7$. Multiply by 2: $x + 3 = 14$, so $x = 11$.

Example 5: solve $\frac{2x - 1}{3} = \frac{x + 4}{2}$.

Multiply both sides by 6 (the common multiple of 3 and 2), or cross-multiply:

$$2(2x - 1) = 3(x + 4)$$ $$4x - 2 = 3x + 12$$ $$x = 14$$

Forming and solving equations

Many exam questions describe a situation. Write an equation, then solve it.

Example 6: the angles in a triangle are $x$, $2x + 10$ and $3x - 10$ degrees. Find $x$.

Angles in a triangle add to $180°$:

$$x + (2x + 10) + (3x - 10) = 180$$ $$6x = 180$$ $$x = 30$$

The angles are $30°$, $70°$ and $80°$, which add to $180°$.

Common mistakes

• Changing only one side of the equation.
• Sign errors when moving terms. Moving $-4$ across means adding 4 to both sides.
• Dividing only some terms. If you divide both sides of $6x + 3 = 15$ by 3, you must divide every term: $2x + 1 = 5$, not $2x + 3 = 5$.
• Not checking. Always substitute your answer back into the original equation.