Starting system:

3x - 4y = -12
5x - 3y = 13

It is often helpful to multiply each equation by a different number to prepare an "ADDITION" which will eliminate one of the 2 variables:

Multiply the top equation by -5
and the bottom equation by 3:

-15x + 20y = +60
15x - 9y = 39

Add the terms in each column: 0x + 11y = 60 + 39
11y = 99
y = 9

Substitute "9" for "y" in the top original equation, to get:

3x - 4(9) = -12
3x = 36 - 12 = 24
x = 8