Example – Solving a word problem using simultaneous equations

Danni has £4.40 in 2p and 5p coins.

She has 100 coins altogether.

Write down 2 equations for this information.

How many of each type of coin does she have?

Solution

So Danni has 20 two-pence coins and 80 fi ve-pence coins.

Check:

t

+

f

= 100 1

Make the coeffi cients of one of the unknowns the same in both equations.

The signs on 5

f

are positive in both equations, so subtract.

There are 100 coins altogether.

Choose letters to represent the unknowns.

Danni has 440 pence altogether.

This matches the 2 and 5 which are in pence.

Divide both sides by 3.

To fi nd the value of

f

, substitute

t

= 20 into equation 1 .

Practising skills

1 Simplify these expressions.

a 5

x

– 9

x

2 Solve these simultaneous equations by adding them fi rst.

a

x

^{+ 2}

y

^{= 0 b 3}

x

⁺

y

^{= 18 c}

x

^{– 5}

y

^{= 2}

x

– 2

y

= 4 2

x

–

y

= 7 3

x

+ 5

y

= –14 d 6

x

^{+ 2}

y

^{= 6 e –}

x

⁺

y

^{= 1 f –2}

x

^{+ 3}

y

^{= 10}

x

^{– 2}

y

^{= 8}

x

⁺

y

^{= –5 2}

x

⁺

y

^{= 6}

3 Solve these simultaneous equations by subtracting them fi rst.

a 2

x

^{– 3}

y

^{= 5 b}

x

^{+ 4}

y

^{= 1 c 3}

x

^{+ 4}

y

^{= 30}

2

x

^{+ 4}

y

^{= 12}

x

^{+ 2}

y

^{= 3}

x

^{+ 4}

y

^{= 26}

d 2

x

– 5

y

= 22 e 5

x

–

y

= 12 f

x

+

y

= 2 3

x

^{– 5}

y

^{= 23 3}

x

^–

y

^{= 8 3}

x

⁺

y

^{= 12}

4 Solve these simultaneous equations. Decide for yourself whether to add them or subtract them.

a 3

x

⁺

y

^{= 7 b 4}

x

^{– 2}

y

^{= 14 c 3}

x

^{+ 4}

y

^{= 32}

2

x

^–

y

^{= 8}

x

^{– 2}

y

^{= 11 3}

x

^{– 2}

y

^{= 2}

d –5

x

^{+ 3}

y

^{= 21 e 6}

x

^–

y

^{= 21 f 2}

x

^{– 4}

y

^{= 16}

5

x

^–2

y

^{= –19 3}

x

⁺

y

^{= 15 3}

x

^{+ 4}

y

^{= 14}

5 Here are two simultaneous equations.

3

x

+ 4

y

= 18 equation 1 5

x

^{– 2}

y

^{= 4 equation 2}

a Explain why you cannot eliminate either variable just by adding or subtracting.

b Which equation should be multiplied, and by which value, to ensure you can eliminate a variable?

c Solve the equations.

6 Solve these simultaneous equations by multiplying one equation fi rst.

a 4

x

^–

y

^{= 19 b 3}

x

^{+ 4}

y

^{= 7 c 5}

x

^–

y

^{= –20}

2

x

^{+ 3}

y

^{= –1}

x

^{+ 2}

y

^{= 1 4}

x

^{+ 3}

y

^{= 3}

d 6

x

^{– 3}

y

^{= 51 e}

x

^{– 5}

y

^{= 31 f 3}

x

^{– 2}

y

^{= –2}

3

x

^{+ 4}

y

^{= 20 4}

x

^{– 2}

y

^{= 16 2}

x

^{– 4}

y

^{= 12}

Developing fl uency

1 Here are two simultaneous equations.

3

x

+ 4

y

= 23 equation 1 2

x

^{+ 3}

y

^{= 16 equation 2}

a In this question you need to multiply each equation so that either

x

or

y

can be eliminated.

Explain 2 ways that this can be done.

b Solve these 2 equations.

2 Solve these simultaneous equations.

a 4

a

+ 3

b

= 18 b 3

x

+ 2

y

= 20 c 3

p

+ 5

q

= –5 3

a

^{+ 4}

b

^{= 17 2}

x

^{+ 5}

y

^{= 17 2}

p

^{– 3}

q

^{= 22}

d 7

x

– 4

y

= 39 e 6

x

– 5

y

= 38 f –4

m

+ 6

n

= –4 3

x

+ 5

y

= 10 5

x

– 3

y

= 27 3

m

+ 2

n

= 29

Unit 4 Solving simultaneous equations by elimination Band h 3 In this diagram the perimeter of the triangle is 118 cm and the perimeter of the

rectangle is 114 cm.

a + 3b a + 2b

2a + b

2a 3a − 4b

a Write two simultaneous equations for

a

^and

b

^.

b Solve them to fi nd the values of

a

^and

b

^.

c Compare the areas of the triangle and the rectangle.

4 In one week, Trevor made 5 journeys to the supermarket and 3 journeys to the park.

He travelled a total of 99 kilometres.

During the next week, he made 2 journeys to the supermarket and 4 to the park.

He travelled a total of 62 kilometres.

a Write down 2 simultaneous equations. Use

s

for the distance to the supermarket and

p

for the distance to the park.

b Solve your equations.

c How far does Trevor travel in a week when he goes 7 times to the park and twice to the supermarket?

5 The price of tickets for a zoo are £

a

for adults and £

c

for children.

Rose buys 2 adult tickets and 3 children’s tickets for £24.

Amanda buys 3 adult tickets and 5 children’s tickets for £38.

a Write down two simultaneous equations.

b Solve them to fi nd the price of each ticket.

c The Singhs have a family outing to the zoo. There are 9 adults and 21 children.

How much do they pay?

6 Reya has these 2 equations.

5

x

^{– 4}

y

^{= 18}

x

– 4

y

= – 6

Her next step is to do a subtraction. She says,

‘My rule is:

Change the signs on the bottom line and go on as if you were adding.’

a Carry out Reya’s rule.

b Explain why Reya’s rule works.

c Solve the equations.

Exam-styleExam-style

Problem solving

1 A banana costs

b

pence and an apple costs

a

^pence.

Robin buys 4 bananas and an apple for 150p.

Marion buys 2 bananas and an apple for 100p.

a Write this information as 2 simultaneous equations.

b Solve the equations.

c Find the cost of 7 bananas and 8 apples.

2 In this diagram, the length and width of the rectangle are

l

cm and

w

cm respectively.

The equal sides of the isosceles triangle are also

l

^cm

and the base

w

^cm.

The perimeter of the rectangle is 32 cm.

The perimeter of the triangle is 26 cm.

a Write this information as two simultaneous equations.

b Find the values of

l

and

w.

3 Karen leaves her 2 dogs and 3 cats at The Pet Hotel for 7 days and it costs her £315.

Malcolm leaves his dog and 2 cats at The Pet Hotel for 7 days and it costs him £175.

One week The Pet Hotel looks after 15 dogs and 20 cats.

How much money do they collect from their customers in that week?

4 Peter makes mountain bikes and sports bikes.

It takes him 4 hours to make a mountain bike and 6 hours to make a sports bike.

Each mountain bike costs him £45 to make and each sports bike costs him £60 to make.

One week Peter made

m

mountain bikes and

s

Sports bikes in 54 hours.

The cost of making these bikes was £570.

a Write this information as a pair of simultaneous equations.

b How many of each type of bike did he make?

5 A mobile phone company charges their pay-as-you-go customers

c

pence per minute for calls and

t

pence for each text message.

Ben is charged £7.30 for making 10 minutes of calls and sending 40 texts.

Alicia is charged £7.25 for making 5 minutes of calls and sending 50 texts.

a Write this information as two simultaneous equations.

b Solve the equations to fi nd the values of

c

^and

t

^.

c Ruby has £20 of credit. She makes 15 minutes of calls and sends 100 texts. How much credit does she have left?

6 a Write an equation for the angles in the triangle.

b Write a second equation connecting the

x

^and

y

^.

Exam-style Higher tier only

The Pet Hotel

Leave your pet in our expert care when you go on holiday.

Exam-styleExam-style

x

°

3

x

°

y

°

Unit 4 Solving simultaneous equations by elimination Band h

Reviewing skills

1 Solve these simultaneous equations.

a 3

x

+ 4

y

= 2 b 2

x

– 3

y

= 2 c

x

– 4

y

= 15 3

x

– 5

y

= 11 5

x

– 3

y

= 14 2

x

+ 4

y

= –6 2 Solve these simultaneous equations.

a

x

^–

y

^{= 7 b 4}

x

^{– 6}

y

^{= 34 c 7}

x

^{+ 4}

y

^{= 27}

5

x

+ 3

y

= 75 3

x

– 3

y

= 21 3

x

+ 2

y

= 11 3 Solve these simultaneous equations.

a 3

x

^{+ 2}

y

^{= 6 b 3}

x

^{– 5}

y

^{= 0 c 4}

x

^{– 5}

y

^{= 17}

2

x

+ 3

y

= 4 4

x

+ 3

y

= 29 3

x

– 2

y

= 18 4 Cedric and Bethany go to a garden centre and buy some trees.

Cedric buys 4 peach trees and 3 apple trees for £75.

Bethany buys 3 peach trees and one apple tree for £45.

a Write this information as a pair of simultaneous equations.

b Solve the equations.

c Find the cost of 3 peach trees and 7 apple trees.

In document Mathematics for Edexcel GCSE - Foundation 2-Higher 1 (2015) (Page 164-169)