How do i solve a system of 2 equations?

This problem is one that can be confusing if not tackled orderly. Let us have an example of a system of 2 equation:

3x + 5y = 13

4x + 7y = 18

First we must choose one of the two and then express one of our variables with respect to the other. Let's take the first one and express y with respect to x:

3x + 5y = 13

5y = 13 - 3x

y = 13/5 - 3x/5

Already it doesn't look very nice, but do not panic! the next step is to use this formula for the next equation. By substituting y in the second equation with this function of x we will have a single equation with only x as an unknown:

4x + 7y = 18

4x + 7*(13/5 - 3x/5) = 18

Now we group the x factors toghether:

4x + 713/5 - 73 x/5 =18

(4 - 7*3/5)x + 713/5 =18

We put the constants on the right hand side and divide the whole equation by the x factor.

(4 - 21/5)x = 18 - 713/5

x = (18 - 91/5) / (4 -21/5)

Now it all comes down to a simple calculation for the numerator and denominator.

numerator = (18*5-91)/5 = -1/5

denominator = (4*5-21)/5 = -1/5

In this case they are equal, so we find x

x = 1

Now we substiture this result in the first equation and we can find the last unknown, y

3x + 5y = 13

3*1 +5y = 13

5y = 13-3

5y = 10

y = 2

The answers are x=1 and y=2. This is the simplest way to solve the 2 equation system and it work for any system. There are some methods that can solve system of more than 2 equations, but that is a topic for another time.