Robinson Tuition
Grade 8 · Higher · free GCSE maths worksheet

Simultaneous equations: one linear and one quadratic

Substituting the linear into the quadratic.

13 original questions, 58 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Solve the simultaneous equations

y = x2

y = x + 6

4 marks
Show answer
x = 3, y = 9 and x = -2, y = 4
2

Solve the simultaneous equations

y = x2 − 3x

y = 2x − 6

4 marks
Show answer
x = 2, y = -2 and x = 3, y = 0
3

Solve the simultaneous equations
y = x2 − 4
3x − y = 6

4 marks
Show answer
x = 1, y = -3 and x = 2, y = 0
4

Solve the simultaneous equations

x2 + y2 = 20

y = 2x

4 marks
Show answer
x = 2, y = 4 and x = -2, y = -4
5

Solve the simultaneous equations

y = x2 + 2x − 1

y = 3x + 5

4 marks
Show answer
x = 3, y = 14 and x = -2, y = -1
6

Solve the simultaneous equations

x2 + y2 = 13

x + y = 5

5 marks
Show answer
x = 2, y = 3 and x = 3, y = 2
7

Solve the simultaneous equations

y = 2x2 − 3

y = 5x

5 marks
Show answer
x = 3, y = 15 and x = -12, y = -52
8

The circle C has equation x2 + y2 = 10

The line L has equation y = x − 2

Find the coordinates of the points where L intersects C.

5 marks
Show answer
(3, 1) and (-1, -3)
9

Solve the simultaneous equations

y = x2

x + y = 12

5 marks
Show answer
x = 3, y = 9 and x = -4, y = 16
10

Solve the simultaneous equations
y = x2 − 2x
y = x + 1
Give your answers in the form a ± √b2 for x and for y, where a and b are integers.

5 marks
Show answer
x = (3 + √13)/2, y = (5 + √13)/2 and x = (3 - √13)/2, y = (5 - √13)/2
11

The curve C has equation y = x2 + 3. The line L has equation y = 2x.
Show that L does not intersect C.

4 marks
Show answer
x2 - 2x + 3 = 0 has discriminant -8 < 0, so no real solutions
12

The curve C has equation y = x2 − 4x + 7

The line L has equation y = 2x − 2

Show that L touches C at exactly one point, and find the coordinates of this point.

4 marks
Show answer
x2 - 6x + 9 = (x - 3)2 = 0 has the repeated root x = 3, so exactly one point of intersection, (3, 4)
13

Solve the simultaneous equations

x2 + y2 = 9

x + 2y = 3

5 marks
Show answer
x = 3, y = 0 and x = -95, y = 125

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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