How do I solve the simultaneous equations 5x - 3y = -1 and 3x + y = 5?

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.