Rukasha Maths

GCSE maths Beta

Simultaneous linear and quadratic equations · Revision guide

Higher · Algebra · 1 page in the PDF

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

Watch out

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