Simultaneous equations
In AS/A-Level Mathematics, there are two methods for solving two linear simultaneous equations.
Elimination
We eliminate either the x or the y terms by multiplying the two equations by suitable numbers and then either adding or subtracting.
For example:
Solve 5x + 7y = 19 (1) and 3x + 2y = 7 (2)
If we multiply equation (1) by 3 and equation (2) by 5 we get:
15x + 21y = 57 (1)x3
15x + 10y = 35 (2)x5
Now subtract:
11y = 22
y = 2
Substitute into (1): 5x + 7(2) = 19
x = 1
Substitution
We make either x or y the subject of one of the equations and then substitute it back into the other equations to find the other variable.
For example:
Solve x + 2y = 13 (1) and 3x + 7y = 44 (2)
From (1) we see that, x = 13 - 2y
Substitute in (2),
3(13 - 2y) + 7y = 44
39 + y = 44
y = 5
Substitute this value in (1),
x = 13 - 2(5)
x = 3
We also need to be able to solve simultaneous equations in which one of the equations is linear and one is not linear.
That's all!