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

Simultaneous equations in two unknowns

About 80 minutesNo calculator75 marks

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Date

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Quick questions to start.

A1
Needed today
Expand and simplify $(2 x - 5)^{2} - x (x - 4)$ .
[2 marks]
Answer
A2
GCSE Higher
Make $y$ the subject of $3 x - 4 y = 10$ .
[2 marks]
Answer $y = \dots\dots$
A3
Last lesson
Use the discriminant to show that $3 x^{2} - 5 x + 4 = 0$ has no real roots.
[2 marks]
A4
From chapter 3
$f (x) = 2 x^{2} - 3$ . Solve $f (x) = 47$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

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 $3 x + 4 y = 13$ and $5 x - 6 y = 47$ .

Hint Make the

y

terms match: multiply the first by

3

and the second by

2

. The signs differ, so add.

B1
Solve the simultaneous equations $5 x + 3 y = 5$ and $2 x - 7 y = - 39$ .
[3 marks]
Answer $x = \dots\dots y = \dots\dots$

Example 2

Solve the simultaneous equations $y = x^{2} - 2 x - 3$ and $y = 2 x + 9$ .

Hint Both expressions equal

y

, so set them equal. Then use the LINEAR equation to find each

y

B2
Solve the simultaneous equations $y = 2 x^{2} + x - 4$ and $y = 3 x + 8$ .
[4 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$

Example 3

The line $y = x + 2$ meets the circle $x^{2} + y^{2} = 20$ at the points $A$ and $B$ . Work out the coordinates of $A$ and $B$ .

Not drawn accurately

Hint Put

x + 2

in place of

y

in the circle's equation. Expand the bracket before you collect terms.

B3
The line $y = 2 x - 5$ meets the circle $x^{2} + y^{2} = 50$ at the points $A$ and $B$ . Work out the coordinates of $A$ and $B$ .
[4 marks]
Not drawn accurately
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$

Example 4

Solve the simultaneous equations $x^{2} - x y + y^{2} = 7$ and $x + y = 1$ .

Hint Write

y = 1 - x

and substitute. Every term must be expanded with care:

- x (1 - x) = - x + x^{2}

B4
Solve the simultaneous equations $2 y^{2} - x y - 2 x^{2} = 13$ and $y - x = 2$ .
[5 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$

Example 5

The line $y = x + k$ is a tangent to the circle $x^{2} + y^{2} = 18$ . Work out the two possible values of $k$ .

Hint A tangent meets the circle at exactly one point, so the quadratic in

x

has a repeated root:

b^{2} - 4 a c = 0

B5
The line $y = 2 x + k$ is a tangent to the circle $x^{2} + y^{2} = 20$ . Work out the two possible values of $k$ .
[4 marks]
Answer $k = \dots\dots$ or $k = \dots\dots$

Practice

Level 2

C1
Four coffees and three muffins cost £13.90. Two coffees and five muffins cost £12.20. Work out the cost of three coffees and two muffins.
[4 marks]
Answer£ $\dots\dots$
C2
The simultaneous equations $a x + b y = 11$ and $b x - a y = 7$ have the solution $x = 3$ , $y = - 1$ . Work out the values of $a$ and $b$ .
[3 marks]
Answer $a = \dots\dots b = \dots\dots$

C3
Show that the line $y = x - 6$ does not meet the curve $y = x^{2} - 3 x$ .
[2 marks]
C4
A rectangle has a perimeter of $38$ cm and an area of $84$ cm². Its length is $x$ cm and its width is $y$ cm, where $x > y$ . Form a pair of simultaneous equations and solve them to work out $x$ and $y$ .
[4 marks]
Not drawn accurately
Answer $x = \dots\dots y = \dots\dots$
C5
Solve the simultaneous equations $\frac{x}{4} + \frac{y}{3} = 4$ and $\frac{x}{2} - \frac{y}{6} = 3$ .
[3 marks]
Answer $x = \dots\dots y = \dots\dots$
C6
Solve the simultaneous equations $x y = 18$ and $y = 2 x - 5$ .
[4 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$

Exam-style questions

AQA exam style

D1
Solve the simultaneous equations
$2 x^{2} - x y + 2 y^{2} = 30 x - y = 3$
Do not use trial and improvement. You must show your working.
[5 marks]
Answer $x = \dots\dots, y = \dots\dots$ and $x = \dots\dots, y = \dots\dots$

D2
Sam says, "The line $y = 4 x - 7$ crosses the curve $y = x^{2} - 2 x + 2$ at two points, because a straight line always meets a $U$ -shaped curve twice."
Show that Sam is wrong, and work out the coordinates of every point where the line meets the curve.
[3 marks]
Answer $(\dots\dots, \dots\dots)$
D3
Two square tiles have sides $x$ cm and $y$ cm, where $x < y$ . The total of their perimeters is $92$ cm. The total of their areas is $277$ cm².
Not drawn accurately
(a)
Show that $x^{2} - 23 x + 126 = 0$ .
[3 marks]
(b)
Hence work out the side of each tile.
[2 marks]
Answer $x = \dots\dots y = \dots\dots$
Total for D3: 5 marks
D4
How many points of intersection do the graphs of $y = x^{2} + 2$ and $y = 2 x$ have?
Circle your answer. $0123$
[1 mark]
D5
The line $y = x + 1$ meets the circle $x^{2} + y^{2} = 13$ at the points $P$ and $Q$ .
Not drawn accurately
(a)
Work out the coordinates of $P$ and $Q$ .
[4 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$
(b)
Work out the length of $P Q$ . Give your answer in the form $a 2$ .
[2 marks]
Answer $\dots\dots 2$
Total for D5: 6 marks
D6
The graphs of $y = x^{2} + p x + q$ and $y = 2 x + 1$ meet at the points $(1, 3)$ and $(4, 9)$ . Work out the values of $p$ and $q$ .
[3 marks]
Answer $p = \dots\dots q = \dots\dots$

Extension

Stretch

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

E1
Solve the simultaneous equations $x + y = 5$ and $x^{3} + y^{3} = 35$ .
[4 marks]
Hint 1 · What to try
Find the value of $x y$ first. Think about how $(x + y)^{3}$ expands.
Hint 2 · The first line
$(x + y)^{3} = x^{3} + y^{3} + 3 x y (x + y)$ , so $125 = 35 + 3 x y \times 5$ .
Hint 3 · The full method
$15 x y = 90$ gives $x y = 6$ . Then $x$ and $y$ are the roots of $t^{2} - 5 t + 6 = 0$ , which are $2$ and $3$ .
Answer $x = \dots\dots, y = \dots\dots$ and $x = \dots\dots, y = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve two linear equations.
I can solve a linear and a quadratic equation as pairs.
I can tell if a line meets, touches or misses a curve.
I can form simultaneous equations from a problem.

Answers and mark scheme

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