One method of solving these is by elimination. We can try to subtract a multiple of one equation from the other to cancel the x's (or y's).
5x - 3y = -1 (A)
3x + y = 5 (B)
In this case, we can add 3*(equation B) to equation A to cancel the y's. We get
14x = 14, so x = 1.
To find y, substitute this value of x into one of the original equations and solve for y.
5 - 3y = -1 (A, sub x = 1)
6 - 3y = 0
3y = 6
y = 2
So the answer is x = 1, y = 2. It's a good idea to check your answer using the other equation (the one you didn't substitute into before).
3x + y = 5 (B)
3 + 2 = 5 (sub x = 1, y = 2)
The last equation is clearly true, so we have in fact found the correct x and y.