AQA GCSE Maths

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GCSEMathsAQATopic Questions2. AlgebraSimultaneous Equations

Simultaneous Equations

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1
Sme Calculator3 marks

Solve the simultaneous equations

4 x space plus space y space equals space 25
x long dash 3 y equals space 16

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2
Sme Calculator3 marks

Solve the simultaneous equations

         2 x minus y space equals space 13 x minus 2 y space equals space 11

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3
Sme Calculator3 marks

Solve the simultaneous equations


   3 x space plus space y space equals space minus 4

   3 x space minus space 4 y space equals space 6

      

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4
Sme Calculator4 marks

Solve the simultaneous equations

2 x space minus 4 y space equals 19
3 x space plus space 5 y space equals space 1

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5
Sme Calculator1 mark

Two straight lines intersect at point A.

q16-paper-3h-june-2018-aqa-gcse-maths

Circle the coordinates of A.

left parenthesis – space 3 over 4 space comma space 3 right parenthesis left parenthesis – 4 comma space 3 right parenthesis left parenthesis – 12 comma space 3 right parenthesis left parenthesis – space 4 over 3 space comma space 3 right parenthesis

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6
Sme Calculator3 marks

Solve the simultaneous equations.

table row cell 2 x space plus space y space end cell equals cell space 18 end cell row cell x space minus space y space end cell equals cell space 6 end cell row blank blank blank end table

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7
Sme Calculator1 mark

Two straight lines intersect at point P.

q13-paper2h-nov2017-aqa-gcse-maths

Circle the coordinates of P.

left parenthesis – 3 comma space – 1 right parenthesis open parentheses negative 1 space comma negative space 1 third close parentheses left parenthesis – 1 comma space – 3 right parenthesis open parentheses negative space 1 third space comma space minus 1 space close parentheses

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8
Sme Calculator3 marks

Solve the simultaneous equations

table row cell 7 x space plus space 2 y space end cell equals cell space 36 space end cell row cell 3 x space plus space 2 y space end cell equals cell space 16 end cell end table

x equals...................... y equals......................

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9
Sme Calculator3 marks

Solve the simultaneous equations

table row cell 4 x space plus space 5 y space end cell equals cell space 4 space end cell row cell 2 x space – space y space end cell equals cell space 9 end cell end table

Show clear algebraic working.

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10
Sme Calculator3 marks

Solve the simultaneous equations

table row cell 4 x space plus space 2 y space end cell equals cell space 9 space space end cell row cell x space – space 4 y space end cell equals cell space 9 end cell end table

Show clear algebraic working.

x space equals space y space equals             

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11
Sme Calculator3 marks

Solve the simultaneous equations

table row cell x space plus space y space end cell equals cell space 15 space end cell row cell 7 x space – space 5 y space end cell equals cell space 3 end cell end table

Show clear algebraic working.

x space equals space....................................................... space space y space equals space.......................................................

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12
Sme Calculator3 marks

Solve the simultaneous equations


table row cell x space plus space 2 y space end cell equals cell space minus 0.5 end cell row cell 3 x space minus space y space end cell equals cell space 16 end cell end table

Show clear algebraic working.

x space equals space............... y space equals space...............

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13
Sme Calculator3 marks

Solve the simultaneous equations

table row cell 3 x space plus space 5 y space end cell equals cell space 6 space end cell row cell 7 x space – space 5 y space end cell equals cell space – 11 end cell end table

Show clear algebraic working.

table row cell x space end cell equals cell space............... end cell row cell y space end cell equals cell space............... end cell end table

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14
Sme Calculator3 marks

Solve.

      4 x space plus space 3 y space equals space 5 space 2 x space plus space 3 y space equals space 1

x = ....................

y = ....................

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1
Sme Calculator4 marks

Solve

      2 x space plus space 3 y space equals space 2 over 3

      3 x space minus space 4 y space equals space 18

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2
Sme Calculator5 marks

Solve the equations 

      x squared space plus space y to the power of 2 space end exponent equals space 36 x space equals space 2 y space plus space 6

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3
Sme Calculator6 marks

Solve the simultaneous equations       x squared space plus space y squared space equals space 9 x space plus space y space equals space 2

Give your answers correct to 2 decimal places.

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4
Sme Calculator5 marks

Solve algebraically the simultaneous equations

      x squared space plus space y squared space equals space 25 y space minus space 3 x space equals space 13

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5
Sme Calculator5 marks

Solve algebraically the simultaneous equations

x squared space plus space y squared space equals space 25 y space minus space 2 x space equals space 5

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6
Sme Calculator4 marks

3 kg of potatoes and 4 kg of carrots have a total cost of 440p.
4 kg of potatoes and 3 kg of carrots have a total cost of 470p.

Work out the total cost of 1 kg of potatoes and 1 kg of carrots.

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7
Sme Calculator4 marks

A cinema sells adult tickets and child tickets.

   The total cost of 3 adult tickets and 1 child ticket is £30
   The total cost of 1 adult ticket and 3 child tickets is £22

Work out the cost of an adult ticket and the cost of a child ticket.

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8
Sme Calculator4 marks

3 teas and 2 coffees have a total cost of £7.80
5 teas and 4 coffees have a total cost of £14.20

Work out the cost of one tea and the cost of one coffee.

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9
Sme Calculator4 marks

Solve the simultaneous equations

table row cell 2 x space plus space 3 y space end cell equals cell space 5 p end cell row cell y space end cell equals cell space 2 x space plus space p end cell end table

where p is a constant.
Give your answers in terms of space p spacein their simplest form.

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10
Sme Calculator4 marks

A linear sequence starts

a space plus space 2 b space space space space space space space space space space space space a space plus space 6 b space space space space space space space space space space space space space a space plus space 10 b space space space space space space space space space space space space space space space........ space space space space space space space space space........

The 2nd term has value 8

The 5th term has value 44

Work out the values of a and b.

a = ......................

b = ......................

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11
Sme Calculator3 marks

Solve the simultaneous equations

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x plus 5 y end cell equals cell 3.1 end cell row cell 6 x plus 3 y end cell equals cell 3.75 end cell end table

Show clear algebraic working.

table row cell x space end cell equals cell space........................... end cell row cell y space end cell equals cell space........................... end cell end table   

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12
Sme Calculator5 marks

The line with equation 2 y space equals space x space plus space 1 intersects the curve with equation 3 y to the power of 2 space end exponent plus space 7 y space plus space 16 space equals space x squared space – space x at the points A and B

Find the coordinates of A and the coordinates of B
Show clear algebraic working.

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13
Sme Calculator6 marks

Solve.

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus y squared end cell equals 34 row y equals cell x plus 2 end cell end table

Show your working.

x space equals space....................... space y space equals space...................... space x space equals space....................... space y space equals space...................... space

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14
Sme Calculator5 marks

The diagrams show the price paid by two groups of people visiting a funfair.

q8-paper4-nov2019-ocr-gcse-maths

Assume each adult pays the same price and each child pays the same price.

Find the price for an adult and the price for a child.

Adult price = £ ........................................
Child price = £ ........................................

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15
Sme Calculator5 marks

Given that

      m open parentheses table row 4 row 1 end table close parentheses space plus space n open parentheses table row 5 row 2 end table close parentheses space equals space open parentheses table row 12 row 6 end table close parentheses

find the value of m and the value of n.

m = .......................

n = ....................... 

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1
Sme Calculator4 marks

Solve the simultaneous equations

   5 x space plus space 2 y space equals space 11 4 x space minus space 3 y space equals space 18

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2
Sme Calculator4 marks

Solve the simultaneous equations

   3 x space plus space 2 y space equals space 4 4 x space plus space 5 y space equals space 17

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3
Sme Calculator4 marks

Solve the simultaneous equations

   4 x space plus space 7 y space equals space 1 3 x space plus 10 y space equals space 15

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4
Sme Calculator4 marks

Solve the simultaneous equations

   3 x space plus space 4 y space equals space 5

   2 x space minus space 3 y space equals space 9

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5
Sme Calculator2 marks

The graphs with equations 3 y space plus space 2 x space equals space 1 half and 2 y space minus space 3 x space equals space minus space 113 over 12 have been drawn on the grid below.

q5-medium-2-11-simulatenous-equation-edexcel-gcse-maths

Using the graphs, find estimates of the solutions of the simultaneous equations

3 y space plus space 2 x space equals 1 half 2 y space minus space 3 x space equals negative space 113 over 12

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6
Sme Calculator4 marks

A pattern is made using identical rectangular tiles.

q6-medium-2-11-simulatenous-equation-edexcel-gcse-maths

Find the total area of the pattern.

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7
Sme Calculator4 marks

Solve the simultaneous equations

table row cell 2 x space plus space 4 y space end cell equals cell space – 9 space end cell row cell 2 y space end cell equals cell space 4 x space – space 7 end cell end table

x equals........................ y equals........................

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8
Sme Calculator3 marks

Here is the graph of 4 x space – space 3 y space equals space 12 for values of x from 0 to 4

q8-paper2h-specimen2015-aqa-gcse-maths

By drawing a second graph on the grid,

work out an approximate solution to the simultaneous equations

4 x space – space 3 y space equals space 12 and  3 x space plus space 2 y space equals space 6

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9
Sme Calculator4 marks

Solve the simultaneous equations     table row cell 2 x space plus space 7 y space end cell equals cell space 17 space end cell row cell 5 x space plus space 3 y space end cell equals cell space – 1 end cell end table

Show clear algebraic working.

table row cell x space end cell equals cell space.................................. end cell row cell y space end cell equals cell space.................................. space end cell end table 

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10
Sme Calculator3 marks

Solve the simultaneous equations

table row cell 5 a space plus space 2 c space end cell equals cell space 10 space end cell row cell 2 a space – space 4 c space end cell equals cell space 7 end cell end table

Show clear algebraic working.


             a space equals .......................................................       
c space equals .......................................................       

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11
Sme Calculator4 marks

Solve the simultaneous equations

table row cell 7 x space – space 2 y space end cell equals cell space 34 space end cell row cell 3 x space plus space 5 y space end cell equals cell space minus 3 end cell end table

Show clear algebraic working.

x space equals.............................................      

y space equals.............................................      

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1
Sme Calculator5 marks

Solve algebraically the simultaneous equations

x squared space minus space 4 y squared space equals space 9 3 x space plus space 4 y space equals space 7

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2
Sme Calculator5 marks

Prove algebraically that the straight line with equation x minus space 2 y space equals space 10 is a tangent to the circle with equation x squared space plus space y squared space equals space 20

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3a
Sme Calculator1 mark

In all the following sequences, after the first two terms, the rule is to add the previous two terms to find the next term.

Write down the next two terms in this sequence.

1      1      2      3      5      8      13      ..........      .........

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3b
Sme Calculator2 marks

Write down the first two terms of this sequence.

.........       .........     3      11      14

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3c
Sme Calculator8 marks
i)
Find the value of d and the value of e.

   2      d   e       10   
[3]
 
ii)
Find the value of x, the value of y and the value of z.

   -33   x   y   z      18
[5]

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4
Sme Calculator5 marks

There are only r red counters and g green counters in a bag.

A counter is taken at random from the bag.

The probability that the counter is green is 3 over 7

The counter is put back in the bag.

2 more red counters and 3 more green counters are put in the bag.
A counter is taken at random from the bag.

The probability that the counter is green is 6 over 13

Find the number of red counters and the number of green counters that were in the bag originally.

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5a
Sme Calculator3 marks

The line y space equals space 3 x space plus space p and the circle x squared space plus space y squared space equals space 53 intersect at points A and B.

p is a positive integer.

Show that the x-coordinates of points A and B satisfy the equation

   10 x squared plus space 6 p x space plus space p squared space – space 53 space equals space 0

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5b
Sme Calculator5 marks

The coordinates of A are (2, 7)

Work out the coordinates of B.
You must show your working.

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6
Sme Calculator3 marks

x space colon space y space equals space 7 space colon space 4

x space plus space y space equals space 88

Work out the value of space x space – space y

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7
Sme Calculator4 marks

A curve has equation y space equals space 4 x squared space plus space 5 x space plus space 3

A line has equation y space equals space x space plus space 2

Show that the curve and the line have exactly one point of intersection.

Do not use a graphical method.

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8
Sme Calculator5 marks

Solve the simultaneous equations

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x squared plus y squared minus x y end cell equals 5 row y equals cell 2 x minus 3 end cell end table                             

Show clear algebraic working.

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9
Sme Calculator5 marks

Solve the simultaneous equations

table row cell x squared space – space 9 y space – space x space end cell equals cell space 2 y squared space – space 12 end cell row cell x space plus space 2 y space – space 1 space end cell equals cell space 0 end cell end table

Show clear algebraic working.

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10
Sme Calculator6 marks

The straight line bold L has equation     x space – space y space equals space 3
The curve bold C has equation 3 x squared – space y squared plus space x y space equals space 9

bold L and bold C intersect at the points P and Q.

Find the coordinates of the midpoint of P Q.
Show clear algebraic working.

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11
Sme Calculator5 marks

Solve the simultaneous equations

table row cell x space – space 6 y space end cell equals cell space 5 space end cell row cell x y space – space 2 y squared space end cell equals cell space 6 end cell end table

Show clear algebraic working.

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12
Sme Calculator5 marks

Solve the simultaneous equations

table row cell y space end cell equals cell space 3 space – space 2 x space end cell row cell x squared space plus space y squared space end cell equals cell space 18 end cell end table

Show clear algebraic working.

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13
Sme Calculator6 marks

The curve with equation x squared – space x space plus space y squared space equals space 10 and the straight line with equation x space – space y space equals space – 4 intersect at the points Aand B.

Work out the exact length of A B.
Show your working clearly and give your answer in the form fraction numerator square root of a over denominator 2 end fraction where a is an integer.

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14
Sme Calculator6 marks

The prices of two phones are in the ratio x : y.

When the prices are both increased by £20, the ratio becomes 5 : 2.
When the prices are both reduced by £5, the ratio becomes 5 : 1.

Express the ratio x : y in its lowest terms.

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15
Sme Calculator4 marks

Li has t toy bricks.
She only has red bricks and blue bricks.

Li picks two bricks, one after the other.

If the first brick she picks is red, the probability that the second brick is red is 2 over 3.

If the first brick she picks is blue, the probability that the second brick is red is 7 over 10.

Calculate the value of t .

t = .........................................................

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16
Sme Calculator6 marks

Solve these simultaneous equations algebraically.

y space space equals space 2 x squared space minus space 7 x space plus space 4

y space equals space 4 x minus space 1

x = ..................... y = .....................
x = ..................... y = .....................

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17
Sme Calculator5 marks

Find the exact coordinates of the two intersections of the line y  = 2x and the circle x2+ y2 = 30

q17-paper6-nov2018-ocr-gcse-maths

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18a
Sme Calculator5 marks

Each week Dan drives two routes, route X and route Y.

One week he drives route X three times and route Y twice.
He drives a total of 134 miles that week.

Another week he drives route X twice and route Y five times.
He drives a total of 203 miles that week.

Find the length of each route.

route X = ........................... miles
route Y = ........................... miles

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18b
Sme Calculator1 mark

State an assumption that has been made in answering part (a).

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