1. Solving harder linear Simultaneous Equations
7x - 4y = 37 1
2x + 3y = -6 2
In this case there is no single multiplier for one equation in order
to make coefficients the same, so we have to multiply both equations
Use the smallest possible multiplier to produce the smallest numbers
To work with 1
x 3
21x - 12y = 111 3 2
x 4
8x + 12y = -24 4
29x = 87 4 3 + 29
x = 3
Check in 1
Substitute in 2
(2 x 3) + 3y = -6
(7 x 3) - (4 x -4) = 37 -6
3y = -12 3
y = -4

2. Solving harder linear Simultaneous Equations
Practice
3x - 2y = 6 1
5x - 3y = 9 2 1
x 3
9x - 6y = 18 3 2
x 2
10x - 6y = 18 4
Subtract 3 4
x = 0
Substitute in 1
(3 x 0) - 2y = 6
-2y = 6 2
y = -3
Check in 2
(5 x 0) – (3 x -3) = 9

3. Solving non-linear Simultaneous Equations
Solve the simultaneous equations:
x2 + y2 = 13
x + 3y = 3
The way to solving these is to convert them into quadratic
equations in the form ax2 + bx +c
Step 1
Rearrange the linear equation because this will be used to
substitute into the quadratic equation.
x = 3 – 3y
Make x the subject of the formula
because making y the subject would
look like this and would be harder
to substitute
Step 2
Substitute x = 3 – 3y into the non-linear equation
(3 - y)2 + y2 = 13
Expand and simplify
(9y2 -18y + 9) + y2 = 13

4. Solving non-linear Simultaneous Equations
Step 3
10y2 -18y + 9 = 13
Rearrange to make the equation = 0
10y2 -18y - 4 = 0
Factorise and solve the quadratic equation
Step 4
(5y+1)(y-2) = 0
y = 2
Step 5
Substitute into the linear equation (easiest)
When
x + 3 x = 3
x =
When y = 2
x + 3 x 2 = 3
x = -3
Check your answers by substitution into the non-linear equation

5. Solving non-linear Simultaneous Equations
Practice
x2 + y2 = 16
y = x - 4
Substitute into the non-linear equation
x2 + (x – 4)2 = 16
Expand and simplify
x2 + (x2 - 8x + 16) = 16
2x2 - 8x + 16 = 16
Rearrange to make the equation = 0
2x2 - 8x = 0
2x(x – 4) = 0
x = 0
x = 4
Substitute into the linear equation
Check the values in
the non linear equation
When x = 0, y = -4
When x = 4, y = 0