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

Let's start by multiplying the first equation by 4 and the second equation by -3 to eliminate the y variable:

4(2x + 3y) = 4(-1)
-3(5x + 4y) = -3(1)

This gives us:

8x + 12y = -4
-15x - 12y = -3

Now, add these two equations together to eliminate the y variable:

8x + 12y - 15x - 12y = -4 - 3
-7x = -7

Divide by -7 on both sides:

x = 1

Now, substitute x = 1 back into one of the original equations to solve for y. Let's use the first equation:

2(1) + 3y = -1
2 + 3y = -1
3y = -3
y = -1

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