AQA Level 2 Further Maths · Level 2 Certificate · Chapter 4: Algebra IV · Lesson 8

Simultaneous equations in three unknowns

About 90 minutesNo calculator76 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/further-maths-simultaneous-equations-in-three-unknowns/

Date

Do Now

The first two questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Solve the simultaneous equations $5 x + 2 y = 19$ and $3 x - 2 y = 5$ .
[2 marks]
Answer $x = \dots\dots y = \dots\dots$
A2
Needed today
$x + 2 y - z = 4$ and $z = 3 x - y$ . Substitute for $z$ and simplify, to give an equation in $x$ and $y$ only.
[2 marks]
Answer $\dots\dots\dots\dots$
A3
Last lesson
Work out the limiting value, as $n \to \infty$ , of the sequence with $n$ th term $\frac{12 n - 5}{3 n + 4}$ .
[2 marks]
Answer
A4
From chapter 3
$f (x) = \frac{2 x + 5}{3}$ . Work out $f^{- 1} (x)$ .
[2 marks]
Answer $f^{- 1} (x) = \dots\dots\dots$

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Solve the simultaneous equations $2 x + 3 y + z = 4$ , $x - 2 y + 3 z = 13$ and $3 x + y - 2 z = - 1$ .

Hint Number the equations. Pick one unknown (here

z

) and remove it from two different pairs of equations. That leaves two equations in

x

and

y

B1
Solve the simultaneous equations $3 x + 2 y - z = 3$ , $2 x - y + 3 z = 0$ and $x + 3 y + 2 z = 15$ .
[5 marks]
Answer $x = \dots\dots y = \dots\dots z = \dots\dots$

Example 2

Solve the simultaneous equations $2 a + 3 c = 13$ , $a - b + 2 c = 9$ and $3 a + 2 b - c = 10$ .

Hint The first equation has no

b

in it already. So eliminate

b

from the other two, and you have two equations in

a

and

c

B2
Solve the simultaneous equations $4 p - r = 16$ , $2 p + 3 q + r = 5$ and $p - q + 2 r = - 6$ .
[4 marks]
Answer $p = \dots\dots q = \dots\dots r = \dots\dots$

Example 3

Solve the simultaneous equations $z = 2 x - y + 1$ , $3 x + 2 y = z + 8$ and $x + y + 2 z = 15$ .

Hint The first equation already gives

z

. Substitute it into the other two equations, then tidy each into the form

a x + b y = c

B3
Solve the simultaneous equations $y = 2 z - x + 1$ , $2 x + 3 z = y - 6$ and $4 x - y + z = - 12$ .
[4 marks]
Answer $x = \dots\dots y = \dots\dots z = \dots\dots$

Example 4

$x = 2$ , $y = - 1$ , $z = 3$ is the solution of the simultaneous equations $a x + b y + cz = 14$ , $b x + cy + a z = 5$ and $c x + a y - b z = 7$ . Work out the values of $a$ , $b$ and $c$ .

Hint Put the values of

x

y

and

z

in. Now

a

b

and

c

are the unknowns, in three ordinary equations.

B4
$x = 3$ , $y = - 2$ , $z = 1$ is the solution of the simultaneous equations $a x + b y + cz = - 5$ , $a x + b z - cy = 9$ and $b x + a y - cz = 12$ . Work out the values of $a$ , $b$ and $c$ .
[5 marks]
Answer $a = \dots\dots b = \dots\dots c = \dots\dots$

Example 5

The curve $y = a x^{2} + b x + c$ passes through the points $(1, 2)$ , $(- 1, 8)$ and $(2, 5)$ . Work out the values of $a$ , $b$ and $c$ .

Hint Each point gives one equation: put its

x

and

y

into

y = a x^{2} + b x + c

B5
The curve $y = a x^{2} + b x + c$ passes through the points $(1, - 4)$ , $(- 2, 17)$ and $(3, 2)$ . Work out the values of $a$ , $b$ and $c$ .
[4 marks]
Answer $a = \dots\dots b = \dots\dots c = \dots\dots$

Practice

Level 2

C1
Solve the simultaneous equations $x + 2 y - z = - 1$ , $3 x - y + 2 z = 13$ and $2 x + 3 y + z = - 2$ by first eliminating $y$ .
[5 marks]
Answer $x = \dots\dots y = \dots\dots z = \dots\dots$
C2
At a theatre, 2 adult tickets, 3 child tickets and 1 senior ticket cost £54. 1 adult ticket, 2 child tickets and 2 senior tickets cost £44. 3 adult tickets, 1 child ticket and 1 senior ticket cost £52. Work out the cost of each type of ticket.
[5 marks]
Answeradult £ $\dots$ child £ $\dots$ senior £ $\dots$

C3
Solve the simultaneous equations $4 x + y + z = 3$ , $2 x - 2 y + 3 z = - 11$ and $6 x + y - 2 z = 10$ .
[5 marks]
Answer $x = \dots\dots y = \dots\dots z = \dots\dots$
C4
The 2nd, 5th and 6th terms of a quadratic sequence are $6$ , $51$ and $74$ . Work out the $n$ th term of the sequence.
[4 marks]
Answer $\dots\dots\dots\dots$

Exam-style questions

AQA exam style

D1
Solve the simultaneous equations $4 a + b - 3 c = - 4$ , $2 a + 3 b + c = - 7$ and $2 a - 2 b + 5 c = 21$ . Do not use trial and improvement. You must show your working.
[5 marks]
Answer $a = \dots\dots b = \dots\dots c = \dots\dots$

D2
Sam says, " $x = 2$ , $y = 1$ , $z = 3$ is the solution of these equations, because it works in the first two."
$x + y + z = 6$ , $2 x + y = 5$ , $3 x - y + 2 z = 4$
(a)
Show that Sam is wrong.
[1 mark]
(b)
Solve the simultaneous equations. You must show your working.
[4 marks]
Answer $x = \dots\dots y = \dots\dots z = \dots\dots$
Total for D2: 5 marks
D3
The curve $y = a x^{2} + b x + c$ passes through the points $(- 1, 10)$ , $(2, 1)$ and $(3, 6)$ .
(a)
Work out the values of $a$ , $b$ and $c$ .
[4 marks]
Answer $a = \dots\dots b = \dots\dots c = \dots\dots$
(b)
Hence show that the line $y = 3 x - 5$ is a tangent to the curve.
[3 marks]
Total for D3: 7 marks
D4
$a + b + c = 7$ , $a + 2 b + 4 c = 21$ , $a + 3 b + 9 c = 43$
(a)
Solve the simultaneous equations.
[4 marks]
Answer $a = \dots\dots b = \dots\dots c = \dots\dots$
(b)
Hence write down an expression for the $n$ th term of the quadratic sequence $7, 21, 43, \dots$
[1 mark]
Answer $\dots\dots\dots\dots$
Total for D4: 5 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A café sells tea at £2, coffee at £3 and cake at £5. A group buys 20 items for £61 in total. They buy at least one of each, and more coffees than teas. Work out every possible number of teas, coffees and cakes.
[5 marks]
Hint 1 · What to try
Call the numbers $t$ , $c$ and $k$ . You have only two equations, so look for whole-number answers.
Hint 2 · The first line
$t + c + k = 20$ and $2 t + 3 c + 5 k = 61$ . Take $2 \times$ the first from the second: $c + 3 k = 21$ .
Hint 3 · The full method
So $c = 21 - 3 k$ and $t = 20 - c - k = 2 k - 1$ . Both at least $1$ means $1 \leq k \leq 6$ ; $c > t$ means $21 - 3 k > 2 k - 1$ , so $k \leq 4$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve three simultaneous equations by eliminating one unknown.
I can rearrange and substitute when an equation is not in the usual form.
I can find unknown coefficients, a quadratic curve or an $n$ th term from three facts.
I can form and solve three equations from a problem in context.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers