This is a system of linear equations; you can tell because the variables have an exponent of 1 (which you don't have to write). These can also be called first-degree or first order expressions.
There are 2 popular methods of solving systems of equations, substitution and elimination. To use substitution, you rewrite one of the equations with one variable alone on one side of the equation. You could do that fairly easily with the second equation, since x does not have a coefficient: x = -2y - 8. You would then substitute "-2y - 8" in for x in the first equation, and solve for x. You then use the value of x to find y.
The second method involves multiplying one or both equations so that one variable can be eliminated by adding, that is, the coefficients are opposites. In this case, we can multiply the 2nd equation by -4 so that the x terms add to zero:
x + 2y = -8 multiply each term by -4 to keep the equation balanced
x(-4) + 2y(-4) = -8(-4)
-4x - 8y = 32 now add the 1st equation
4x + 5y = -8
0 - 3y = 24
-3y = 24 divide both sides by -3
y = -8 use -8 for y to find x in either equation
x + 2(-8) = -8
x - 16 = -8
x = 8
The solution is (8, -8). You can double check that it works in the first equation as well:
4(8) + 5(-8) = -8
32 - 40 = -8 it works, so the solution is (8, -8).