GCSE: Further Simultaneous
Equations
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
www.drfrostmaths.com
Last modified: 31st August 2015
Starter
Solve the following simultaneous (linear) equations.
2x + 3y = 8
4x – y = -5
x = -0.5
?
y=3
RECAP :: Equation of a circle
y
The equation of this circle is:
x2 + y2 =? 25
5
-5
5
-5
x
! The equation of a circle with
centre at the origin and radius r
is:
x2 + y2 = r2
Quickfire Circles
1
3
1
-1
-1
2 = 16
x2 + y?
6
10
8
-8
x2 + y2 = 64
10
-10
4
-4
-4
x2 + y 2 = 9
8
?
3
-3
x2 + y?2 = 1
-8
?
-3
4
-10
2 = 100
x2 + y?
?
-6
6
-6
x2 + y2 = 36
Motivation
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
STEP 1: Rearrange linear equation to
make x or y the subject.
10
x=y+2
?
STEP 2: Substitute into quadratic and
solve.
10
10
-2
x
(y + 2)2 + y2 = 100
y2 + 4y + 4 + y2 = 100
2y2 + 4y – 96 = 0?
y2 + 2y – 48 = 0
(y + 8)(y – 6) = 0
y = -8 or y = 6
10
STEP 3: Use either equation to find
the values of the other variable.
When y = -8, x =?-6
When y = 6, x = 8
Your Go
y = x2 – 3x + 4
y–x=1
STEP 1: Rearrange linear equation to
make x or y the subject.
STEP 3: Use either equation to find
the values of the other variable.
y=1+x
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) + 4
4 = 32 – (3 x 3) +?4
C
C
Exercises
1
y = x2 + 7x – 2
y = 2x – 8
x = -3, y = -14
x = -2, y =?-12
7
2
x2 + y2 = 8
y=x+4
x = -2, y =?+2
8
3
y = x2
y=x+2
x = -3, y = -14
x = -2, y =?-12
4
5
6
x2 + y2 = 5
x – 2y = 5
x = 1, y = ?-2
y = x2 – x – 2
x + 2y = 11
x = 3, y = 4
x = -5/2, y?= 27/4
y = x2 – 2x – 2
x = 2y + 1
x = 3, y = 1
x = -1/2, y?= -3/4
x+y=1
x2 + y 2 = 1
x = 1, y = 0
x = 0, ?
y=1
x2 – y2 = 15
2x + 3y = 5
x = -8, y = 7
x = 4, ?
y = -1
9
x2 – y2 = 15
2x + 3y = 5
x = 5, y = -3
x = 191/59, y = -255/59
10
y = x2 – 3x
x=y–9
N
?
x = 2 – √13, y = 11 – √13
X = 2 + √13, y = 11 + √13
x2 – 4y + 7 = 0
y2 – 6z + 14 = 0
z2 – 2x – 7 = 0
[Source: BMO]
?
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)