Academic Skills Advice
Simultaneous Equations
Simultaneous equations have values of 𝑥 and 𝑦 that work for all equations. If you had 2
equations and they were represented on a graph this would be the point where the two
lines meet (i.e. the intersection of the lines).
The equations:
𝑥+𝑦 =5
and 𝑦 = 4𝑥 − 5 are simultaneous because the
values 𝑥 = 2 and 𝑦 = 3 work in both equations (try them and see).
We will look at 2 possible ways of solving simultaneous equations, by elimination and by
substitution:
Solving by Elimination:
1) Write the equations in the same order. (line up the 𝑥’s and 𝑦’s)
2) Make the numbers in front of the 𝑥′𝑠 OR the 𝑦′𝑠 the same. (whichever seems easier)
3) Same signs:
subtract one equation from the other. Resulting in a new equation.
Different signs: add the equations together.
4) Solve the new equation to find 𝑥 or 𝑦 (see lesson 2 1st teaching sheet for more help).
5) Substitute back into one of the original equations to find the other letter.
Examples (elimination):

Solve simultaneously:
Number the equations as you
go along to make it easier.
𝟐𝒚 + 𝒙 = 𝟏𝟏
𝒚 + 𝟑𝒙 = 𝟖
(𝑦’s and 𝑥’s already lined up)
2𝑦 + 𝑥 = 11
𝑦 + 3𝑥 = 8
(1)
(2)
Make the numbers in front of 𝑥 the same:
multiply every term in (1) by 3:
equation (2) stays the same:
6𝑦 + 3𝑥 = 33
𝑦 + 3𝑥 = 8
(3)
Subtract because 𝑥′𝑠 are the same sign (both +):
5𝑦
Same order:
= 25
(Equation (3) – equation (2)):
Solve to find 𝑦:
𝑦=5
Substitute into either (1) or (2) to find 𝑥.
𝑦 + 3𝑥 = 8
5 + 3𝑥 = 8
𝑥=1
Replace 𝑦 with 5:
Solve to find 𝑥:
It doesn’t matter which equation you choose to substitute into as they will both give you the
same answer. Once you have found the answer, check it in the other equation to make
sure it works.
We have found that the values: 𝒙 = 𝟏 and 𝒚 = 𝟓 will work in both equations. We could
check them both as follows:
Equation (1):
Equation (2):
2𝑦 + 𝑥 = 11
𝑦 + 3𝑥 = 8
© H Jackson 2011 / ACADEMIC SKILLS
2(5) + 1 = 11
5 + 3(1) = 8
10 + 1 = 11 
5+3=8

1

Solve simultaneously:
It’s easy to make the numbers
in front of 𝑦 the same, as 2 and
3 both go into 6.
𝟑𝒙 − 𝟐𝒚 = 𝟖
𝟖𝒙 = 𝟑𝟖 − 𝟑𝒚
3𝑥 − 2𝑦 = 8
8𝑥 + 3𝑦 = 38
(1)
(2)
Make the numbers in front of 𝑦 the same:
multiply every term in (1) by 3:
multiply every term in (2) by 2:
9𝑥 − 6𝑥 = 24
16𝑥 + 6𝑦 = 76
(3)
(4)
Add because 𝑦′𝑠 are different signs:
25𝑥
Same order:
= 100
(Equation (3) + equation (4)):
Solve to find 𝑥:
𝑥=4
Substitute into either (1) or (2) to find 𝑥.
3𝑥 − 2𝑦 = 8
3(4) − 2𝑦 = 8
𝑦=2
Replace 𝑥 with 4:
Solve to find 𝑦:
Checks:
Equation (1):
Equation (2):

3𝑥 − 2𝑦 = 8
8𝑥 + 3𝑦 = 38
Solve simultaneously:
3(4) − 2(2) = 8
8(4) + 3(2) = 38
12 − 4 = 8 
32 + 6 = 38 
𝟑𝒙 − 𝟒𝒚 = 𝟏𝟕
𝟓𝒙 = 𝟏𝟗 + 𝟐𝒚
3𝑥 − 4𝑦 = 17
5𝑥 − 2𝑦 = 19
(1)
(2)
Make the numbers in front of 𝑦 the same:
equation (2) stays the same:
multiply every term in (2) by 2:
3𝑥 − 4𝑦 = 17
10𝑥 − 4𝑦 = 38
(3)
Subtract because 𝑦′𝑠 are the same signs:
7𝑥
Same order:
= 21
(Equation (3) - equation (1)):
Solve to find 𝑥:
𝑥=3
Substitute into either (1) or (2) to find 𝑦.
3𝑥 − 4𝑦 = 17
3(3) − 4𝑦 = 17
𝑦 = −2
Replace 𝑥 with 4:
Solve to find 𝑦:
Checks:
Equation (1):
Equation (2):
3𝑥 − 4𝑦 = 17
5𝑥 − 2𝑦 = 19
© H Jackson 2011 / ACADEMIC SKILLS
3(3) − 4(−2) = 17 9 + 8 = 17 
5(3) − 2(−2) = 19 15 + 4 = 19 
2
Solving by Substitution:
1) Rearrange one of the equations (if necessary) to make either 𝒙 or 𝒚 the subject (see
lesson 2 1st teaching sheet for more help).
2) Substitute either 𝑥 or 𝑦 into the other equation.
3) Solve the new equation to find 𝑥 or 𝑦.
4) Substitute back into your rearranged equation to find the value of the other letter.
Examples (substitution):

Solve simultaneously:
Rearrange equation 2:
𝟐𝒚 + 𝒙 = 𝟏𝟏
𝒚 + 𝟑𝒙 = 𝟖
(1)
(2)
𝑦 = −3𝑥 + 8
(to make 𝑦 the subject)
Substitute 𝑦 into equation 1:
(Replace 𝑦 with −3𝑥 + 8)
2𝑦 + 𝑥 = 11
2(−3𝑥 + 8) + 𝑥 = 11
Tidy up and solve for 𝑥:
−6𝑥 + 16 + 𝑥 = 11
−5𝑥 = −5
𝑥=1
Substitute back in original to find 𝑦:
𝑦 = −3𝑥 + 8
𝑦 = −3(1) + 8
𝑦=5
Checks:
Equation (1):
Equation (2):

2𝑦 + 𝑥 = 11
𝑦 + 3𝑥 = 8
Solve simultaneously:
Rearrange equation 1:
You can choose to
rearrange whichever you
prefer so just choose
whichever seems easier.
2(5) + 1 = 11
5 + 3(1) = 8
10 + 1 = 11 
5+3=8

𝟐𝒙 − 𝟒𝒚 = −𝟏𝟔
𝟑𝒙 + 𝟓𝒚 = 𝟗
(1)
(2)
𝑥 = −8 + 2𝑦
(to make 𝑥 the subject)
Substitute 𝑥 into equation 2:
(Replace 𝑥 with −8 + 2𝑦)
3𝑥 + 5𝑦 = 9
3(−8 + 2𝑦) + 5𝑦 = 9
Tidy up and solve for 𝑦:
−24 + 6𝑦 + 5𝑦 = 9
11𝑦 = 33
𝑦=3
Substitute back in original to find 𝑥:
𝑥 = −8 + 2𝑦
𝑥 = −8 + 2(3)
𝑥 = −2
Checks:
Equation (1):
Equation (2):
2𝑥 − 4𝑦 = −16
3𝑥 + 5𝑦 = 9
© H Jackson 2011 / ACADEMIC SKILLS
2(−2) − 4(3) = −16
3(−2) + 5(3) = 9
−4 − 12 = −16
−6 + 15 = 9


3
Simultaneous Equation with 3 Unknowns:
To be able to solve simultaneous equations with 3 unknowns you must have 3 equations.
You can use the elimination method first to eliminate one of the letters then solve as
normal. It is quite a long process and there are other methods that you may learn in future.
Examples (3 unknowns):

Solve simultaneously:
Same order:
Make the numbers in front of 𝑦 the same:
equation (1) x2:
equation (2) x6:
equation (3) x3:
𝟐𝒙 + 𝟑𝒚 − 𝟐𝒛 = 𝟑
𝒚 − 𝒛 = 𝟑𝒙 − 𝟖
𝟕𝒛 = 𝟓𝒙 − 𝟐𝒚 + 𝟑𝟏
(1)
(2)
(3)
2𝑥 + 3𝑦 − 2𝑧 = 3
−3𝑥 + 𝑦 − 𝑧 = −8
−5𝑥 + 2𝑦 + 7𝑧 = 31
(1)
(2)
(3)
4𝑥 + 6𝑦 − 4𝑧 = 6
−18𝑥 + 6𝑦 − 6𝑧 = −48
−15𝑥 + 6𝑦 + 21𝑧 = 93
(4)
(5)
(6)
Now we look at pairs of equations to eliminate the 𝑦’s.
Equation (4) – equation (5) =
Equation (6) – equation (5) =
22𝑥 + 2𝑧 = 54
3𝑥 + 27𝑧 = 141
(7)
(8)
(n.b. you can use any pairs of equations – just choose which you find the easiest)
Now we have just 2 equations and can choose elimination or substitution to solve them.
e.g. Rearranging equation (7) gives:
𝑧 = −11𝑥 + 27
Substituting into equation (8) gives:
3𝑥 + 27(−11𝑥 + 27) = 141
Tidy up & solve:
3𝑥 − 297𝑥 + 729 = 141
−294𝑥 = −588
𝑥=2
Substitute into equation (7):
22(2) + 2𝑧 = 54
44 + 2𝑧 = 54
2𝑧 = 54 − 44
2𝑧 = 10
𝑧=5
Substitute both into equation (1):
(to find 𝑦)
2(2) + 3𝑦 − 2(5) = 3
4 + 3𝑦 − 10 = 3
3𝑦 = 3 − 4 + 10
3𝑦 = 9
𝑦=3
(Finally check that the values work in all 3 equations.)
© H Jackson 2011 / ACADEMIC SKILLS
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