How do you solve simultaneous equations where both equations are linear?

A method that always works is to express one variable as a function of another, and then substitute that expression into the other equation.

Example:

[1] x+y = 5

[2] 2x+3y = 8

[1] x = 5-y

[1] -> [2] 2(5-y)+3y = 8

10-2y+3y = 8

y = -2

x = 5-(-2) = 7

Solution: (7, -2)

Note: There are other ways of solving these kinds of equations, depending on the coefficients, but this method always works.