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

Let's use the method of substitution.

First, solve the second equation for y:
2x - y = -5
y = 2x + 5

Now substitute y = 2x + 5 into the first equation:
5x + 3(2x + 5) = 4
5x + 6x + 15 = 4
11x + 15 = 4
11x = -11
x = -1

Now that we have found the value of x, we can substitute it back into one of the original equations to find y:
2(-1) - y = -5
-2 - y = -5
y = 3

Therefore, the solution to the system of equations is x = -1 and y = 3.