Academic Skills Advice
Algebra Refresher Sheet 3
Simultaneous Equations:
Solving simultaneous equations means finding the value of 𝑥 and 𝑦 that works for both
equations. If the equations were represented on a graph this would be the point where the
two lines meet.
The equations:
𝑥+𝑦 =5
and
𝑦 = 4𝑥 − 5
Are simultaneous because the values 𝑥 = 2 and 𝑦 = 3 work in both equations (check
them).
Solving by Elimination:
1) Make the numbers in front of the 𝑥′𝑠 OR the 𝑦′𝑠 the same.
2) Same signs:
subtract one equation from the other.
Different signs: add the equations together.
3) Solve as linear equation to find 𝑥 or 𝑦 (see Algebra 2 for more help).
4) Substitute back into one of the original equations to find the other letter.
Examples:

Solve simultaneously: 2𝑦 + 𝑥 = 11
𝑦 + 3𝑥 = 8
(equation 1)
(equation 2)
First multiply every term of ‘equation 1’ by 3 to make the numbers in front of 𝑥 the same:
6𝑦 + 3𝑥 = 33
(multiply every term by 3)
𝑦 + 3𝑥 = 8
(equation 2 stays the same)
5𝑦
= 25
(equation 1 take away equation 2)
We subtract the equations because the 𝑥′𝑠 are the same sign (both +)
Now we know:
𝑦=5
We substitute this value into either of the original equations to find 𝑥.
5 + 3𝑥 = 8
(substitute in equation 2)
𝑥=1
(solve to find 𝑥)

Solve simultaneously: 3𝑥 − 2𝑦 = 8
8𝑥 + 3𝑦 = 38
(equation 1)
(equation 2)
Multiply ‘equation 1’ by 3 and ‘equation 2’ by 2 so that the numbers in front of 𝑦 both
become 6.
9𝑥 − 6𝑦 = 24
equation 1 (x3)
16𝑥 + 6𝑦 = 76
equation 2 (x2)
25𝑥
= 100
This time add the equations because the signs in front of the 𝑦 ′ 𝑠 are different (+ and -)
Substitute into original to find:
© H Jackson 2008 / Academic Skills
𝑥=4
𝑦=2
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Solving by Substitution:
1) Rearrange one of the equations (if necessary) to make either 𝑥 or 𝑦 the subject (see
Formulae Refresher Sheet).
2) Substitute either 𝑥 or 𝑦 into the other equation.
3) Solve as linear equation to find 𝑥 or 𝑦.
4) Substitute back into one of the original equations to find the value of the other letter.
Examples (Substitution):

Solve simultaneously:
Rearrange equation 2:
Substitute 𝑦 into equation 1:
2𝑦 + 𝑥 = 11
𝑦 + 3𝑥 = 8
(equation 1)
(equation 2)
𝑦 = −3𝑥 + 8
(we have made 𝑦 the subject)
2(−3𝑥 + 8) + 𝑥 = 11
Notice that the 𝑦 in equation 1 has been substituted with −3𝑥 + 8 (the value we found
when we rearranged equation 2).
Tidy up and solve for 𝑥:
−6𝑥 + 16 + 𝑥 = 11
−5𝑥 = −5
𝑥=1
Substitute back in original equation to find:
𝑦=5
For both of the above methods always check that both of your values (𝒙 and 𝒚) work
in both of the equations.
© H Jackson 2008 / Academic Skills
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Practice Questions:
Solve the following simultaneous equations using whichever method you prefer:
5𝑥 + 4𝑦 = 17
2𝑥 + 4𝑦 = 8
2𝑥 + 2𝑦 = 16
𝑥 + 2𝑦 = 9
5𝑥 − 𝑦 = 3
3𝑥 − 𝑦 = 1
2𝑦 + 𝑥 = 14
5𝑦 + 3𝑥 = 37
3𝑦 + 3𝑥 = 12
4𝑦 − 8𝑥 = −8
4𝑦 + 3𝑥 = 16
9𝑦 + 5𝑥 = 29
−𝑦 + 2𝑥 = 1
3𝑦 − 5𝑥 = 0
𝑦 = 2𝑥 − 9
2𝑦 + 𝑥 = 17
2𝑥 + 𝑦 = 8
3𝑥 − 2𝑦 = 5
𝑥 + 5𝑦 = 8
3𝑥 + 2𝑦 = 11
4𝑥 + 2𝑦 = 8
𝑥 + 3𝑦 = 2
3𝑥 + 2𝑦 = 7
5𝑥 + 4𝑦 = 12
7𝑥 + 3𝑦 = 16
2𝑥 + 9𝑦 = 29
𝑥 + 8𝑦 = 6
3𝑥 + 6𝑦 = 9
6𝑥 + 7𝑦 = 13
3𝑥 − 2𝑦 = 1
8𝑥 + 3𝑦 = 27
2𝑥 − 5𝑦 = 1
11𝑥 + 9𝑦 = 6
5𝑥 + 3𝑦 = 6
5𝑥 + 2𝑦 = 6
3𝑥 − 10𝑦 = 26
4𝑦 + 3𝑥 = 5
9𝑦 − 4𝑥 = 22
4𝑦 + 3𝑥 = 1
5𝑦 + 10𝑥 = 20
−3𝑦 + 2𝑥 = −17
8𝑦 + 5𝑥 = 66
© H Jackson 2008 / Academic Skills
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