Algebra – Forming & Solving Equations

Linear equations, equations with fractions, simultaneous equations and quadratic equations.

Algebra is heavily tested in all three units. You must be fluent at forming equations from worded problems and solving them accurately.

Solving linear equations

**Golden rule:** Do the same operation to both sides. - Add/subtract the same value to both sides - Multiply/divide both sides by the same value - Expand brackets first if needed - Collect like terms

Worked example

Solve: 5x − 7 = 3x + 11

Solution

5x − 7 = 3x + 11 Subtract 3x: 2x − 7 = 11 Add 7: 2x = 18 **x = 9**

Equations with brackets

Always expand brackets carefully, then collect like terms.

Worked example

Solve: 3(2x + 5) = 4(x − 1) + 7

Solution

Expand: 6x + 15 = 4x − 4 + 7 6x + 15 = 4x + 3 Subtract 4x: 2x + 15 = 3 Subtract 15: 2x = −12 **x = −6**

Simultaneous equations (Higher)

**Elimination:** Make coefficients of one variable the same, then add or subtract. **Substitution:** Rearrange one equation, then substitute into the other.

Worked example

Solve: 2x + 3y = 13 4x − y = 5

Solution

(1) 2x + 3y = 13 (2) 4x − y = 5 Multiply (2) by 3: (3) 12x − 3y = 15 Add (1)+(3): 14x = 28 → **x = 2** Into (2): 8 − y = 5 → **y = 3**

Quadratic equations (Higher)

Methods: factorisation, quadratic formula x = [-b ± √(b²−4ac)]/(2a), completing the square. Always rearrange to = 0 first.

Worked example

Solve by factorisation: x² − 5x − 14 = 0

Solution

Numbers that multiply to −14 and add to −5: −7 and +2 (x − 7)(x + 2) = 0 **x = 7** or **x = −2**

Practice questions

Try these without looking at the answers first.

Q1.Solve: 7x + 3 = 4x + 18

Show answer

x = 5

Q2.Solve: 2(3x − 4) = 5x + 7

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x = 15

Q3.Solve simultaneously: 3x + 2y = 16, x − y = 2

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x = 4, y = 2

Q4.Solve by factorisation: x² + 2x − 15 = 0

Show answer

x = 3 or x = −5

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