Simultaneous Equations
Revision Notes
Two linear equations
If two equations are true for the same values i.e. true simultaneously, we can add or
subtract them to give a third equation that is also true for the same values. This method of
solving simultaneous equations is called the elimination method.
e.g.
3x + y = 9
5x – y = 7
You need to have the same number in front of either the x or the y before adding or
subtracting the equations. Whichever has the same number in each equation is the
variable(letter) you will eliminate. In this example there are the same amount of y,
therefore this is the variable that will be eliminated. Once the variable to be eliminated has
been decided, you need to decide whether the equations need to be added or subtracted to
eliminate it. If the signs of the variable to be eliminated are the same then you need to
subtract the equations. If the signs are different then the equations need to be added. In
this example the signs are different so the equations are added.
3x + y = 9
+
5x – y = 7
8x
= 16
x=2
Adding the two equations eliminated the y terms and gave us a single equation in x.
Solving this equation gave us the solution x = 2.
To find the value of y when x = 2 substitute this value into one of the equations.
3x + y = 9
5x – y = 7
Substituting x = 2 into the first equation gives us -
(3 × 2) + y = 9
6+y=9
y=3
The solutions therefore for this set of simultaneous equations are x = 2 and y = 3.
Sometimes you will need to multiply one or both of the equations before you can eliminate
one of the variables because there isn’t the same amount of x’s in both equations or y’s in
both equations.
4x – y = 29
e.g.
3x + 2y = 19
For this example you can multiply all of the first equation by two. This will then give us the
same number of y’s in both equations.
Then solve as in the first example.
8x – 2y = 58
+ 3x + 2y = 19
11x
= 77
x = 11
Substitute into the first equation
4x – y = 29
(4x11) – y = 29
y = 15
The solutions therefore for this set of simultaneous equations are x = 11 and y = 15.
Two simultaneous equations can also be solved by substituting one equation into the other.
This method is called the substitution method.
e.g.
y = 2x – 3
2x + 3y = 23
Substitute equation 1 into equation 2
2x + 3(2x - 3) = 23
2x + 6x – 9 = 23
8x – 9 = 23
8x = 32
x=4
1
2
Substitute the x value into equation 1
y = 2x – 3
y = (2 x 4) – 3
y=5
The solutions therefore for this set of simultaneous equations are x = 4 and y = 5.
I would suggest only using the substitution method to solve two linear equations when one
of the equations is either y = something or x = something. Use the elimination method if
neither of the equations are in this form.
One linear and one quadratic
When one equation is linear and one equation is quadratic you need to use the substitution
method.
e.g.
2
y=x +1
1
y=x+3
2
Substitute equation 1 into equation 2
2
x +1=x+3
Rearrange and then factorise to get the solutions for x
2
x -x–2=0
(x + 1)(x – 2) = 0
Therefore
x + 1= 0  x = -1
or
x–2=0x=2
Then substitute each x value into the original linear equation
When x = -1  y = x + 3 = -1 + 3 = 2
x = -1 and y = 2
When x = 2  y = x + 3 = 2 + 3 = 5
x = 2 and y = 5
One linear and one circle
When one equation is linear and one equation is a circle you need to use the substitution
method.
e.g.
y=x+1
2
1
2
x + y = 13 2
Substitute equation 1 into equation 2 and then simplify
2
2
x + (x + 1) = 13
2
2x + 2x + 1= 13
2
2x + 2x – 12 = 0
2
x +x–6=0
Factorise to get the solutions for x
2
x +x–6=0
(x + 3)(x – 2) = 0
Therefore
x + 3= 0  x = -3
or
x–2=0x=2
Then substitute each x value into the original linear equation
When x = -3  y = x + 1 = -3 + 1 = -2
x = -3 and y = -2
When x = 2  y = x + 1 = 2 + 1 = 3
x = 2 and y = 3
Key Points


In the eliminaton method, you need to have the same number in front of either the x
or the y before adding or subtracting the equations. Whichever has the same
number in each equation is the variable(letter) you will eliminate.
Use the substitution method if the simultaneous equations are a linear and either a
quadratic or a circle.
Exam Questions
Q1.
Solve the simultaneous equations
3x – 2y = 7
7x + 2y = 13
......................................................................................................................................
(Total for Question is 3 marks)
Q2.
Solve the simultaneous equations
3x + 10y = 7
x – 4y = 6
x=.......................
y=.......................
(Total for Question is 3 marks)
Q3.
Solve the simultaneous equations
4x + 7y = 1
3x + 10y = 15
x=......................
y=......................
(Total for Question is 4 marks)
Q4.
Solve the simultaneous equations
5x + 2y = 11
4x – 3y = 18
x=..............
y=..............
(Total for Question is 4 marks)
Q5.
Solve the simultaneous equations x2 + y2 = 9
x+y=2
Give your answers correct to 2 decimal places.
x=...............y=...............
or x = . . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . .
(Total for Question is 6 marks)
Q6.
Solve the simultaneous equations
x2 + y2 = 25
y = 2x + 5
x = . . . . . . . . . . . . . . and y = . . . . . . . . . . . . . .
or
x = . . . . . . . . . . . . . . and y = . . . . . . . . . . . . . .
(Total for Question is 6 marks)
Mock Exam Question