1. Non - Graphical Solution of Simultaneous Equations
Equation: x + 5 = 10
We can solve easily – one equation, one variable.
Can we find a single solution to this equation?
y = 3x + 2
No, because with 2 variables it states the equation of a straight line
A linear equation
In order to solve for 2 variables we are finding where two
lines intersect.

2. Non - Graphical Solution of Simultaneous Equations
Solve this pair of simultaneous equations:
x + 2y = 7
and
x = y - 8
First, write one equation directly under the other, keeping the ‘=‘
Vertically in line and number the equations
x + 2y = 7 1
x = y - 8 2
Rearrange the equations if necessary so that both unknown
variables are on the same side of the equation.
If the equation looks different give it a new number
x + 2y = 7 1
x – y = - 8 3

3. Non - Graphical Solution of Simultaneous Equations
x + 2y = 7 1
x – y = - 8 3
We can now use the 2 equations together
If we subtract equation 3 from equation 1 1 3 -
3y = 15
y = 5
Now we have found one variable we can use this by
substitution into one of the original equations to find the other variable
Substitute in 1
x + 2 x 5 = 7
x + 10 = 7
x = -3
We can check to see whether we have the right answers by
substituting the values in the other original equation.
-3 = 5 - 8

4. Non - Graphical Solution of Simultaneous Equations
Practice
3x – y = 10 1
x + y = -2 2 1 2 +
4x = 8
x = 2
Substitute in 2
2 + y = -2
y = -4
Check in 1
(3 x 2) – (-4) = 10

5. Non - Graphical Solution of Simultaneous Equations
What happens when there is not a common coefficient of the
a variable between the two equations?
5x + 3y = 27 1 2
2x + y = 10
because there is not a common coefficient we need to make one
by multiplying one of the equations.
Always look for the easiest way
Multiplying by 3 would be easiest because then we would have 3y 2 3
6x + 3y = 30
We could subtract from
But that would leave us with
negative numbers…. Messy 1 3
Now subtract from 3
6x + 3y = 30
5x + 3y = 27 1
Check in 1 3 1 -
x = 3
5 x x 4 = 27 1
Substitute in 2
2 x 3 + y = 10
y = 4

6. Non - Graphical Solution of Simultaneous Equations
Practice
6x + 2y = -10 1
2x + 4y = -10 2 1
x 2
12x + 4y = -20 3
Subtract 2 3
12x + 4y = -20 3
2x + 4y = -10 2 2 3 -
10x = -10
x = -1
Substitute in 1
Check in 2
(6 x -1) + 2y = -10
(2 x -1) + (4 x -2) = -10
-6 + 2y = -10
2y = -4
y = -2