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!