Rukasha Maths

GCSE maths Beta

Simultaneous linear and quadratic equations · Higher notes

Higher · Algebra · AQA references A19

A simultaneous solution must satisfy both equations at the same time. On a graph, each solution is an intersection. When one equation is linear, rearrange it to express one variable in terms of the other, then substitute that expression into the quadratic equation. This reduces the problem to one variable. A line and a quadratic may meet twice, once or not at all. After solving, substitute each x-value back separately so you report correctly matched coordinate pairs.

Learning objectives

  • Substitute a linear expression into a quadratic equation.
  • Find and check every matching coordinate pair.

Key ideas

At an intersection, both equations hold for the same coordinate pair.

Substitute the whole expression, using brackets whenever it is squared.

Solve the resulting quadratic; there may be two, one or no real solutions.

Recover the matching second coordinate for every first coordinate. Check both equations.

Worked examples

Example 1

Solve the line and quadratic simultaneously.

y=x+1,y=x2−1y=x+1,\qquad y=x^2-1
x+1=x2−1⇒(x−2)(x+1)=0x+1=x^2-1\quad\Rightarrow\quad(x-2)(x+1)=0
x=2⇒y=3,x=−1⇒y=0x=2\Rightarrow y=3,\qquad x=-1\Rightarrow y=0

Answer

(x,y)=(2,3)or(−1,0)(x,y)=(2,3)\quad\text{or}\quad(-1,0)

Example 2

Solve the horizontal line and circle simultaneously.

y=3,x2+y2=25⇒x2=16y=3,\quad x^2+y^2=25\quad\Rightarrow\quad x^2=16
x=±4x=\pm4

Answer

(x,y)=(4,3)or(−4,3)(x,y)=(4,3)\quad\text{or}\quad(-4,3)

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Solve simultaneous linear and quadratic equations.

AO2 · Reasoning

Explain why the intersection points satisfy both equations.

AO3 · Problem solving

Form a pair of equations from geometric or numerical conditions and interpret solutions.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Do not mix coordinates from different solutions. A positive square can come from either a positive or negative number.

Try it yourself

Attempt each question before opening its answer.

Check 1: Solve y = 3 and x² + y² = 25.

Answer: (4, 3) or (-4, 3)

Substitute y = 3 to get x² = 16, giving both signs.

Check 2: How many intersections do y = x² and y = -1 have?

Answer: None

A real square is non-negative, so it cannot equal -1.

Explain it in your own words

  • Explain what a simultaneous solution means on a graph.
  • Show the substituted equation when y = 2x + 1 is put into x² + y² = 10.
  • Write why matching coordinate pairs matters.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.