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A LevelMaths: PureOCRExam Questions2. Algebra & Functions2.3 Simultaneous Equations

Simultaneous Equations (OCR A Level Maths: Pure)

Exam Questions

3 hours33 questions
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1
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Solve the simultaneous equations

x plus y equals 8
x minus y equals 4

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2
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Solve the simultaneous equations

2 x plus 5 y equals 3
6 x minus 10 y equals 34

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3a
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Show that the equation 4 x plus 2 y minus 6 equals 0 can be written as y equals 3 minus 2 x.

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3b
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By substituting the result from part (a) into the equation 3 x plus 2 y minus 1 equals 1 spacesolve the equations

4 x plus 2 y minus 6 equals 0
3 x plus 2 y minus 1 equals 1

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4
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Solve the simultaneous equations

5 x minus 2 y minus 16 equals 0
3 x plus 7 y plus 15 equals 0

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5
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Substitute y equals x plus 3 spaceinto the equation 2 x squared minus y squared equals 5 x plus 3 in order to solve the equations simultaneously.

Clearly state which values of x correspond to which values of y from your solutions.

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6
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Solve the simultaneous equations

y equals 2 x minus 1
x squared plus y squared minus 2 equals 0

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7
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Solve the simultaneous equations

y equals 2 x plus 3
4 x minus 3 y plus 4 equals negative 1

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8
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Solve the simultaneous equations

y minus x minus 1 equals 0
left parenthesis 2 x plus 1 right parenthesis squared minus 3 y squared plus 3 x minus 10 equals 0

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9
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Solve the simultaneous equations

x plus 3 y minus 1 equals 0
x squared plus 9 y equals 2 minus y squared

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1
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Use elimination to solve the simultaneous equations

7 x plus 4 y equals 17
3 x minus 2 y equals 11

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2
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Use substitution to solve the simultaneous equations

2 x minus 5 y equals 4
x plus y equals negative 5

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3a
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By eliminating y from the equations

3 x squared plus 4 y equals 83
3 x plus 2 y equals negative 11

show that x squared minus 2 x minus 35 equals 0.

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3b
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Hence solve the simultaneous equations

3 x squared plus 4 y equals 83
3 x plus 2 y equals negative 11

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4a
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By eliminating y from the equations

x squared plus 10 x plus y squared equals negative 20
y equals 2 x plus 10

show that x squared plus 10 x plus 24 equals 0.

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4b
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Hence solve the simultaneous equations

x squared plus 10 x plus y squared equals negative 20
y equals 2 x plus 10

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5a
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By eliminating y from the equations

x squared minus 8 y equals negative 40
3 x plus 2 y equals 4

show that x squared plus 12 x plus 24 equals 0.

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5b
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Hence solve the simultaneous equations

x squared minus 8 y equals negative 40
3 x plus 2 y equals 4

giving x and y in the form a plus-or-minus b square root of 3, where a and b are integers.

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6a
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4 x minus k y equals 13
3 x plus 2 k y equals 7

are simultaneous equations, where k is a constant.

Show that x equals 3.

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6b
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Find an expression for y in terms of the constant k.

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6c
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Given that space y equals 3, find the value of k.

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7a
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x squared minus 2 y equals 10
2 x minus y equals k

are simultaneous equations, where k is a constant.

By eliminating y from the equations show that x squared minus 4 x plus 2 left parenthesis k minus 5 right parenthesis equals 0.

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7b
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By considering the discriminant of x squared minus 4 x plus 2 left parenthesis k minus 5 right parenthesis equals 0 find the value of k for which the simultaneous equations have only one solution.

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7c
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Find the solution to the simultaneous equations for the value of k that you found in part (b).

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8
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You are asked to advise a client on which parcel delivery service to use to deliver parcels of differing sizes.  Linear Deliveries Inc. charges a flat rate of £2.25 per parcel, plus 40p times the mass of the parcel in kilograms.  Square Deal Delivery Solutions charges a flat rate of £4 per parcel, plus 1p times the square of the parcel’s mass in kilograms.  Under what circumstances would you advise your client to use each of the two delivery services?  Be sure to show clear mathematical justifications for your answer.

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1
Sme Calculator4 marks

Use elimination to solve the simultaneous equations

3 x plus 4 y equals negative 13
2 x minus 3 y equals 14

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2
Sme Calculator4 marks

Use substitution to solve the simultaneous equations

4 y minus 5 x equals negative 2
3 x plus 2 y equals negative 12

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