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5 Simultaneous equations
This unit will help you to set up and solve different types of simultaneous
equations.
AO1 Fluency check
1 Simplify
b4y − y
a −2x + 3x c 3x − x d−2y + (−2y)
2 Solve
a 3x = 9
x=
c x + 3 = 10
b2y = −8
y=
x =
d3 + 2y = −4
y =
3 Substitute y = 2 into 3x + 2y = 16 and solve to find the value of x.
x=
Number sense
4
Use the operations to complete each equation.
x + y = 10
×2
×3
×5
2x + 2y =
x+
y = 50
3x + 3y =
Key points
Simultaneous equations are two (or more)
equations that involve two (or more) letters.
These
1
Solving simultaneous equations means finding
the value of all the letters.
skills boosts will help you to write and solve simultaneous equations.
Subtracting
to eliminate a
variable
2
Adding to
eliminate a
variable
3
Multiplying
an equation
first
4
Setting up
simultaneous
equations
You might have already done some work on simultaneous equations. Before starting the first skills
boost, rate your confidence with these questions.
1
x + 2y = 7
x+y=4
2
3x + 5y = 32
x − 5y = −16
3
4
5x + 2y = 12
4x + y = 9
3 shirts and 2 hats
cost £61.
2 shirts and 2 hats
cost £46.
How much is a shirt?
How much is a hat?
How
confident
are you?
Unit 5 Simultaneous equations     31
Skills boost
1
Subtracting to eliminate a variable
To solve a pair of simultaneous equations, add or subtract them to eliminate one of the variables.
When there are two identical terms with the same sign, subtract one equation from the other.
Guided practice
Worked
exam
question
Solve
2x + 3y = 9
2x +   y = 7
Label one equation A and one equation B.
2x + 3y = 9 A
2x +   y = 7 B
2y = 2 so y = 1
A
Subtract B from A.
3y −   y =
B
Solve the equation to find the value of y.
y=
,y=
x=
2 a Solve
4x – 6y = – 18
4x + 2y = 22
,y=
x=
y y y
x
x
y
7
6
x=3
Substitute y =
into A.
=9
2x +
2x =
x=3
Solution x = 3, y = 1
2x + 4y = 14
2x + y = 5
x
0
A − B
1 a Solve
x
9
1
Check your solutions work
for equation B.
2 × 3 + 1 = 7 ✓
bSolve
x + 3y = 10
x − y = 2
x =
,y=
c Solve
x + 5y = 7
3x + 5y = 11
x =
bSolve
5x + 2y = 17
6x + 2y = 20
x =
,y=
,y=
c Solve
2x + 3y = 5
2x + 4y = 8
x =
,y=
Hint Simultaneous equations can have negative solutions.
Exam-style question
3 Solve
x + 3y = 11
4 a 3x – 2y = 10
x – 2y = 2
x=
Reflect
,y=
,y=
x=
2x + 3y = 16
b2x – y = 8
5x – y = 23
x =
,y=
c 2x + y = 13
5x + y = 4
x =
How do you decide which equation to subtract from the other?
32     Unit 5 Simultaneous equations
(3 marks)
,y=
Skills boost
2
Adding to eliminate a variable
When two terms have the same coefficient and variable but different signs, add the equations to
eliminate the variable.
Guided practice
Solve
2x – y = 11
 x + y = 10
Subtracting does not eliminate a variable:
2x −   y = 11
A
  x +   y = 10
B
Label one equation A and one equation B.
2x – y = 11 A
 x + y = 10 B
A − B :   x − 2y = 1
You still have x and y.
The coefficients of y are the same and the signs
are different, you can add the equations to
eliminate the y variable.
Add A and B and solve to find x.
A + B
3x + 0 =
3x =
x=
Substitute x = 7 into B.
+ y = 10
y=
Solution x = 7, y = 3
Check your solution works
for equation A.
2 × 7 − 3 = 11 ✓
1 a Solve
bSolve
x + 2y = 7
3x − 2y = 13
3x − y = 5
x + y = 7
,y=
x=
x =
,y=
c Solve
2x + 3y = 26
x – 3y = 4
x =
,y=
Exam-style questions
2 Solve
–x + 3y = 1
x – y = 3
x=
,y=
(3 marks)
x=
,y=
(3 marks)
3 Solve
x + 2y = 12
3x – 2y = 12
4 Solve
a 2x – 3y = −15
b–4x + y = −9
c –x + 3y = 1
4x + 3y = 33 4x – 3y = −5 x = 3 + y
x=
,y=
x=
,y=
x =
,y=
Hint Rearrange the second
equation so that x and y are on
the same side of the equals sign.
Reflect
How do you decide whether to add or subtract the equations?
Unit 5 Simultaneous equations     33
Skills boost
3
Multiplying an equation first
If the x or y coefficients are not equal, multiply one or both of the equations so either the two
x terms or the two y terms have the same coefficient.
Guided practice
Worked
exam
question
Solve
2x + y = −1
x + 4y = 10
Label one equation A and one equation B.
2x +   y = –1 A
  x + 4y = 10 B
Multiply A by 4.
x+
y = –4
Subtract B from
8x + 4y
  x + 4y
7x + 0
x
The coefficient of y in equation A is 1.
1 × 4 = 4, so multiplying A by 4 makes the
y coefficients in A and B equal.
A ×4
4 × A and solve to find x.
= –4 4 × A
−
= 10
B
= –14
= –2
You could multiply B by 2 instead so that both
equations have the same x coefficient.
Substitute x = –2 into A.
(2 × –2) + y = –1
+ y = –1
y=
Solution x = –2, y = 3
1 Solve
3x + 2y = 17
x + y = 7
x=
Check your solutions work
for equation B.
−2 + (4 × 3) = 10 ✓
Hint Either multiply B by 3 (to make the x coefficients
equal) or by 2 (to make the y coefficients equal).
A
B
,y=
2 Solve
a 3x − y = 10
bx + 2y = −1
x + 2y = 8 −3x + y = 10
x=
,y=
x =
,y=
c 2x − 3y = −4
x + y = −7
x =
,y=
3 Solve
a x + 2y = 0
b5x − y = 37
c 2x + −4y = 0
3x − y = −14 x − 3y = 13 x + 2y = 12
x=
,y=
x =
Hint Is it easier to multiply
the first equation by 3 or
the second equation by 2?
34     Unit 5 Simultaneous equations
,y=
x =
,y=
Skills boost
4 Solve
a 10x − 5y = –55 b2x + 5y = 1
c 3x + 2y = 36
x − 2y = –19 –x – y = –5 2x – y = 17
x=
,y=
,y=
x =
,y=
Hint Multiply both equations so two terms have the
5 Solve
2x + 3y = 7
5x + 2y = 1
x=
x =
same coefficient.
You could multiply A by 5 and B by 2.
A
B
,y=
6 Solve
a 4x + 3y = –5
b3x + 7y = –4
c 6x + 2y = 10
3x + 5y = –1 2x + 5y = –2 5x + 10y = 100
x=
,y=
x =
,y=
x =
,y=
7 Solve
a x + 2y = 3
b5x − y = 5
c x + 5y = −1
4x + 2y = 9 10x + 2y = −2 8x − 20y = 28
x=
,y=
x =
,y=
x =
,y=
Hint Solutions to simultaneous equations can be fractions.
8 Solve
a 8x + 2y = 2
b2x − 5y = −14
c −4x + 3y = 13
−4x − 3y = 3 −3x + 2y = −1 6x − 7y = −32
x=
,y=
x =
,y=
x =
,y=
Exam-style question
9 Solve
4x + 2y = 5
8x − 3y = 24
x=
Reflect
,y=
(3 marks)
How did you decide which equation to multiply?
Unit 5 Simultaneous equations     35
Skills boost
4
Setting up simultaneous equations
Guided practice
2 cookies and 3 sandwiches cost £8.
5 cookies and 1 sandwich cost £7.
Find the cost of a a cookie
ba sandwich.
Write the equations using x and y.
£8
Let x = cost of a cookie
and y = cost of a sandwich.
A
x
y
y
y
2
3
cookies sandwiches
Label the equations A and B.
A
2x + 3y = 8
5x +   y = 7
B
Multiply B by 3.
y=
15x +
x
£7
B
x
x
x
x
5
cookies
3× B
Subtract A from 3 × B and solve to find x.
15x + 3y = 21
3× B
A −
 2x + 3y = 8
13x + 0 =
x=
x
y
1
sandwich
1 cookie costs £x
2 cookies cost £2x
Substitute x = 1 into equation A and solve to find y.
a A cookie costs £
b A sandwich costs £
1 At a café 3 lemonades and 1 cola cost £5. 2 lemonades and 4 colas cost £10.
Work out the cost of
a a lemonade ba cola. 2 4 juices and 1 cake cost £3. 2 juices and 3 cakes cost £4. Work out the cost of
a a juice ba cake. 3 Tickets for 2 adults and 3 students cost £38. The cost for 5 adults and 2 students is £62.
Work out the price of
a an adult ticket ba student ticket. 4 Entry for 2 adults and 5 children costs £41. Entry for 3 adults and 2 children costs £34.
Work out the price of
a adult entry bchild entry. Exam-style question
5 4 pens and 3 notebooks cost £17. 6 pens and 2 notebooks cost £15.50.
Find the cost of
a a pen Reflect
ba notebook.
Do you always need to use x and y when setting up equations?
36     Unit 5 Simultaneous equations
(5 marks)
Get back
on track
Practise the methods
Answer this question to check where to start.
Check up
What is the simplest way to start solving these equations?
a  x + 5y = 11
b 5x + y = 7
A
B
Multiply a by 7
C
Multiply b by 5
D
Multiply a by 5
If you ticked B or C finish solving the
equations. Then go to Q2.
Subtract b from a
If you ticked A or D go to Q1 for more
practice.
1 Solve
a 2x + y = 32
3x + y = 42 x=
,y=
bx + 2y = 9
3x − 2y = 11
x =
,y=
c 5x + 2y = 21
−5x + y = 3
x =
,y=
2 Solve
a 2x + y = 10
x − 2y = −5 x=
,y=
b3x − 2y = 18
x + 4y = 13
x =
,y=
c 24x + 3y = 0
2x + y = −2
x =
3 Solve
a 2x + 4y = 0
3x + 2y = 12
x=
,y=
bx + 2y = 0
3x − y = −14
x =
,y=
,y=
c 8x − y = 7
2x − 3y = 10
x =
,y=
4 Solve
a 3x + 7y = 15
5x + 2y = −4
x=
,y=
c 3x + 5y = 3
b 3x + 6y = 3
4x − 4y = −512x + 10y = −2
x =
,y=
x =
,y=
5 At a fair the cost of a ride for 1 adult and 3 children is £11.
The cost for 2 adults and 2 children is also £11.
Work out the cost of
a an adult ticket
ba child ticket.
Exam-style question
6 Solve
2x + 5y = −19
3x − 4y = 29
x=
,y=
(3 marks)
Unit 5 Simultaneous equations     37
Get back
on track
Problem-solve!
1 At the cinema tickets for 2 adults and 3 children cost £33.
Tickets for 1 adult and 1 child cost £14.
Write and solve a pair of simultaneous equations to find the cost of
a an adult ticket
ba child ticket.
2 x and y are two numbers.
Their sum x + y equals 16.
Their difference x − y equals 6.
What are the values of x and y? x =
,y=
3 Find the two numbers whose difference is 15 and whose sum is 35.
,
Exam-style questions
4 Solve these simultaneous questions
4x = y + 7
10x – 2y = 15
x=
,y=
(4 marks)
h=
,b=
(3 marks)
5 Jane’s wages are calculated using the formula
Wages = number of hours, n × hourly rate, h + bonus, b
W = nh + b
When she works 30 hours her wage is £330.
When she works 34 hours her wage is £370.
What is Jane’s hourly rate and what is her bonus?
6 Write equations and solve to find the cost of these drinks and snacks.
a 3 teas and 1 coffee cost £7.
2 teas and 4 coffees cost £13.  
b4 milkshakes and 1 cookie cost £8.20.
2 milkshakes and 3 cookies cost £7.10.  
7 Three first class stamps and one second class stamp cost £2.47.
One first class stamp and two second class stamps cost £1.74.
What is the price difference between first and second class stamps?
8 Why is it not possible to solve these simultaneous equations?
A 3x – 8y = 10
B6x – 16y = 20 Now that you have completed this unit, how confident do you feel?
1
Subtracting
to eliminate a
variable
2
Adding to
eliminate a
variable
38     Unit 5 Simultaneous equations
3
Multiplying an
equation first
4
Setting up
simultaneous
equations