To solve this system of equations, we can use either the substitution method or the elimination method.

Let's use the elimination method to solve this system:

1) x + 8y = -6
2) 5x - 2y = 12

To eliminate one of the variables, let's multiply the first equation by 5 and the second equation by 1:
1) 5(x + 8y) = 5(-6)
2) 1(5x - 2y) = 1(12)

This gives us:
1) 5x + 40y = -30
2) 5x - 2y = 12

Now, we can subtract the second equation from the first:
(5x + 40y) - (5x - 2y) = -30 - 12
5x + 40y - 5x + 2y = -42
42y = -42
y = -1

Now that we have found the value of y, we can substitute it back into one of the original equations to solve for x. Let's use the first equation:
x + 8(-1) = -6
x - 8 = -6
x = 2

Therefore, the solution to the system of equations is x = 2 and y = -1.