I can't seem to fully understand how to get through the whole 
process to the answer.

I'll put in some reasons and a few extra steps which I hope 
will help you.

                   

In the book the problem states as follows:

 let y=0           5y-3x =15
		 5(0)-3x =15  In place of y he wrote (0)
                    0-3x = 15 He multiplied 5 by (0) and got 0
        	     -3x = 15  He didn't need to write the 0	
                   =   He divided both sides by -3   
		       x = -5  (15 divided by -3 is -5) 

let x=0		5y-3x = 15
	      5y-3(0) = 15  In place of x he wrote (0)
                 5y-0 = 15  He multiplied 3 by (0) and got 0 
		   5y = 15  He didn't need to write the 0
                   =   He divided both sides by 5
                    y = 3   (15 divided by 5 is 3).

let y=6		5(6)-3x = 15 In place of y he wrote (6)  
		  30-3x = 15 Multiplied 5 by (6) and got 30 
               30-3x-30 = 15-30  Subtracted 30 from both sides (or added -30 to both sides) 
                   0-3x = -15  Wrote 0 for 30-30 on the left.
                               On the right 15-30 = -15
         	    -3x = -15  Didn't need to write the 0
                  =  Divided both sides by -3   
                      x = 5  (-15 divided by -3 is +5).
		      
Maybe this will help.

Edwin