solution: Name the points $A(5,3)$, $B(6,8)$, $C(7,4)$, and $D(x,y)$ and sketch the first three.  We find that there are three possible locations for $D$ (see figure).  Only the one to the right has an $x$-coordinate greater than 7.  Since $AC$ is parallel to $BD$ and equal in length to it, $D$ is two units to the right and one unit up from $B$, just as $C$ is two units to the right and one unit up from $A$.  Therefore, the coordinates of $D$ are $(8,9)$, and $x+y=8+9=\boxed{17}$.
  
  [asy]
  size(5cm);
  import graph;
  defaultpen(linewidth(0.7)+fontsize(10));
  dotfactor=5;
  real x = 7;
  pair A=(5,3), B=(6,8), C=(7,4), D1=(8,9), D2=(4,7),D3=(6,-1);
  pair[] dots = {A,B,C};
  dot(dots);
  xaxis(-2,10,Ticks(" ",1.0,begin=false,end=false,NoZero),Arrows(4));
  yaxis(-2,10,Ticks(" ",1.0,begin=false,end=false,NoZero),Arrows(4));
  draw(A--C--D1--B--cycle);//linetype("8 8"));
  draw(A--D3--C);
  draw(A--C--B--D2--cycle);//,linetype("1 2 3 1"));
  label("$A(5,3)$",A,SW);
  label("$B(6,8)$",B,NW);
  label("$C(7,4)$",C,E);
  dot(D1,UnFill);
  dot(D2,UnFill);
  dot(D3,UnFill);[/asy]
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  The coordinates of a parallelogram are (5, 3), (6, 8), (7, 4) and $(x, y)$ and $x > 7$. What is the value of $x + y$?
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 17
extra_info.question: The coordinates of a parallelogram are (5, 3), (6, 8), (7, 4) and $(x, y)$ and $x > 7$. What is the value of $x + y$?
extra_info.split: test