A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 3: Equations and inequalities · Lesson 1

Simultaneous equations, linear and quadratic

72 marks

Hints online, answers free with an account:
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Date

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Quick questions to start.

A1
Needed today
Solve $2 x + y = 13$ and $x - 3 y = - 4$ .
[2 marks]
Answer $x = \dots\dots$ $y = \dots\dots$
A2
Needed today
Expand and simplify $(5 - 2 y)^{2}$ .
[2 marks]
Answer
A3
Last lesson
Write $3 + 8 x - 2 x^{2}$ in the form $q - p (x - r)^{2}$ .
[2 marks]
Answer
A4
From chapter 2
Work out the discriminant of $2 x^{2} - 7 x + 8$ . How many real roots does $2 x^{2} - 7 x + 8 = 0$ have?
[2 marks]
Answer

A-Level

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

Example 1

Solve the simultaneous equations $3 x + 4 y = 5$ and $5 x - 6 y = 21$ .

Hint Make the

y

-terms the same size: multiply the first by

3

and the second by

2

, then add.

B1
Solve the simultaneous equations $4 x - 3 y = 23$ and $6 x + 5 y = 6$ .
[3 marks]
Answer $x = \dots\dots$ $y = \dots\dots$

Example 2

Solve the simultaneous equations $x + 3 y = - 2$ and $x^{2} + x y = 8$ .

Hint Write

x = - 2 - 3 y

and put it, in brackets, into the second equation. Find each

y

, then the

x

that goes with it.

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

Example 3

The line $y = 3 x + k$ meets the curve $y = x^{2} - x + 2$ at exactly one point. Find $k$ , and the coordinates of that point.

Hint Put the two expressions for

y

equal. One meeting point means the quadratic has one root, so its discriminant is

0

B3
The line $y = k - 3 x$ , where $k$ is a positive constant, meets the curve $x y = 27$ at exactly one point. Find $k$ , and the coordinates of that point.
[4 marks]
Answer $k = \dots\dots$ $(\dots\dots, \dots\dots)$

A-Level

C1
Solve the simultaneous equations $3 (x + 2 y) = 2 x - 1$ and $\frac{x - y}{2} = y + 4$ .
[4 marks]
Answer $x = \dots\dots$ $y = \dots\dots$

C2
The sketch shows the curve $y = 2 x^{2} - 3 x - 4$ and the line $y = x + 2$ . They meet at $A$ and $B$ . Find the coordinates of $A$ and $B$ .
[4 marks]
Not drawn accurately
Answer $A (\dots\dots, \dots\dots)$ $B (\dots\dots, \dots\dots)$
C3
The simultaneous equations $y = k x - 1$ and $x^{2} + x y = 10$ , where $k$ is a constant, have one solution $x = 2$ , $y = p$ . Find $p$ and $k$ , and find the other solution.
[5 marks]
Answer $p = \dots\dots$ $k = \dots\dots$ $x = \dots\dots, y = \dots\dots$

Edexcel exam style

Stretch

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve linear simultaneous equations.
I can solve a linear and a quadratic pair.
I can pair each $x$ with its own $y$ .
I can find $k$ when a line touches a curve.

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