I'm doing system of equations and I have a problem X=4+y,Y=3+x.

7 for both X and Y
 D**k+Butt=GAYLORD

I do not think there is an answer to this.

x = 4 +y    substitute this into the second equation:  y = 3 + x  

    then  y = 3 + (4+y) 

   then y = y +7       

  then   0 = 7     INVALID !     

Are you sure you wrote the equations in your question correctly??

x = 4 + y    →   x - y  = 4   (1)

y = 3 + x   →   -x - y  = 3  (2)

Add (1)  and (2)   and  we get

0 = 7

Therefore, as EP said........no solutions exist