Simultaneous equations

Linear simultaneous equations

Two equations with two unknowns (usually x and y). We need to find the pair of values that satisfies both equations at the same time.

Method 1: Elimination

Add or subtract the equations to eliminate one variable.

Example: Solve 3x + 2y = 12 and x − 2y = 4. Adding: (3x + x) + (2y − 2y) = 12 + 4 → 4x = 16 → x = 4. Substitute: 4 − 2y = 4 → y = 0. Solution: x = 4, y = 0.

When coefficients don't match: multiply one or both equations to make coefficients equal before eliminating.

Example: 2x + 3y = 13 and 5x − y = 7. Multiply second by 3: 15x − 3y = 21. Add: 17x = 34 → x = 2. Substitute: 4 + 3y = 13 → y = 3.

Method 2: Substitution

Rearrange one equation to express x (or y) in terms of the other, then substitute into the second equation.

Example: y = 2x − 1 and 3x + y = 9. Substitute y: 3x + (2x − 1) = 9 → 5x = 10 → x = 2, y = 3.

Graphical interpretation

The solution to a pair of linear simultaneous equations is the point of intersection of the two lines. CCEA Paper 2 sometimes asks you to draw both lines and read off the intersection.

Linear and quadratic simultaneous equations (Higher)

One linear equation and one quadratic. Use substitution (always from the linear into the quadratic).

Example: y = x + 3 and y = x² − 1. Substitute: x + 3 = x² − 1 → x² − x − 4 = 0. Use quadratic formula: x = (1 ± √17)/2 → x ≈ 2.56 or x ≈ −1.56. Find y values by substituting back.

The number of solutions = number of intersection points of the line and curve:

• Two solutions → line crosses the curve.
• One solution → line is tangent to the curve.
• No solutions → line and curve do not meet.

CCEA context

CCEA Paper 1: linear simultaneous equations (typically elimination or substitution with integer answers). Paper 2: mixed linear/quadratic requiring the quadratic formula, or a worded problem requiring you to set up the equations first.