1. GCSE: Quadratic Simultaneous Equations
Skipton Girls’ High School

2. 2x + 3y = 8 4x – y = -5 Starter x = -0.5 y = 3 ?
Solve the following simultaneous (linear) equations.
2x + 3y = 8
4x – y = -5 ?
x = -0.5
y = 3

3. RECAP :: Equation of a circley
The equation of this circle is:
x2 + y2 = 25 ? 5
! The equation of a circle with centre at the origin and radius r is:
x2 + y2 = r2 x -5 5 -5

4. Quickfire Circles ? ? ? ? ? ? x2 + y2 = 1 x2 + y2 = 9 x2 + y2 = 163 4 -1 1 -3 3 -4 4 -1 -3 -4 ? ?
x2 + y2 = 1
x2 + y2 = 9
x2 + y2 = 16 ? 8 10 ? 6 -8 8
-10 10 -6 6 -8
-10 -6 ?
x2 + y2 = 64
x2 + y2 = 100
x2 + y2 = 36

5. Motivation ? ? ? y 10 x = y + 2 (y + 2)2 + y2 = 100
Given a circle and a line, we may wish to find the point(s) at which the circle and line intersect.
How could we do this algebraically? y 10 ?
STEP 1: Rearrange linear equation to make x or y the subject.
x = y + 2
x – y = 2
x2 + y2 = 100 ?
STEP 2: Substitute into quadratic and solve.
(y + 2)2 + y2 = 100
y2 + 4y y2 = 100
2y2 + 4y – 96 = 0
y2 + 2y – 48 = 0
(y + 8)(y – 6) = 0
y = -8 or y = 6 x 10 10 -2 10 ?
STEP 3: Use either equation to find the values of the other variable.
When y = -8, x = -6
When y = 6, x = 8

6. y = x2 – 3x + 4 y – x = 1 Your Go ? ? ? ? y = 1 + x
STEP 1: Rearrange linear equation to make x or y the subject.
y = 1 + x
STEP 3: Use either equation to find the values of the other variable.
When x = 1, y = 2
When x = 3, y = 4 ? ?
STEP 2: Substitute into quadratic and solve for one variable.
1 + x = x2 – 3x + 4
x2 – 4x + 3 = 0
(x – 1)(x – 3) = 0
x = 1 or x = 3
STEP 4 (OPTIONAL): Check that your pairs of values work.
2 = 12 – (3 x 1) C
4 = 32 – (3 x 3) C ? ?

7. Exercises ? ? ? ? ? ? ? ? ? ? ? x + y = 1 x2 + y2 = 1 x2 – y2 = 15
y = x2 + 7x – 2
y = 2x – 8
x2 + y2 = 8
y = x + 4
y = x2
y = x + 2
x2 + y2 = 5
x – 2y = 5
y = x2 – x – 2
x + 2y = 11
y = x2 – 2x – 2
x = 2y + 1 ?
x = -3, y = -14
x = -2, y = -12 1
x + y = 1
x2 + y2 = 1
x2 – y2 = 15
2x + 3y = 5
y = x2 – 3x
x = y – 9
x2 – 4y + 7 = 0
y2 – 6z + 14 = 0
z2 – 2x – 7 = 0
[Source: BMO] 7 ?
x = 1, y = 0
x = 0, y = 1 2
x = -2, y = +2 ?
x = -8, y = 7
x = 4, y = -1 ? 8 3 ?
x = -3, y = -14
x = -2, y = -12 9 ?
x = 5, y = -3
x = 191/59, y = -255/59 4 10 ?
x = 1, y = -2 ?
x = 2 – √13, y = 11 – √13
X = 2 + √13, y = 11 + √13 ?
x = 3, y = 4
x = -5/2, y = 27/4 5 N ?
x = 1, y = 2,
z = 3
(Add equations, then complete the squares – you’ll end up with a sum of squares which must each be 0)
x = 3, y = 1
x = -1/2, y = -3/4 ? 6