The coordinates of a parallelogram are (5, 3), (6, 8), (7, 4) and $(x, y)$ and $x > 7$. What is the value of $x + y$?
To solve this problem, let's use the fact that the diagonals of a parallelogram bisect each other. Let's pick (5, 3) and (7, 4). Their midpoint is $(6, \frac{7}{2})$. Now, let's find the midpoint of the other diagonal. The other diagonal has endpoints (6, 8) and $(x, y)$, and let's call its midpoint $(\frac{x+6}{2}, \frac{y+8}{2})$. Now, since the diagonals bisect each other, those midpoints are equal. So we have the system of equations $\frac{x+6}{2} = 6$ and $\frac{y+8}{2} = \frac{7}{2}$. The first equation simplifies to $x+6=12$, so $x=6$. Right. So, we have to pick a different pair of points. Let's take (5, 3) and (6, 8). Now we find the midpoint of the other diagonal. The endpoints of the other diagonal are (5, 3) and $(x, y)$. Let's call the midpoint $(\frac{x+5}{2}, \frac{y+3}{2})$. Now, since the diagonals bisect each other, we have the system of equations $\frac{x+5}{2} = \frac{13}{2}$ and $\frac{y+3}{2} = 6$. The first equation simplifies to $x+5 = 13$, so $x = 8$. And that satisfies $x>7$. The second equation simplifies to $y+3=12$, so $y=9$. So the point $(x, y)$ is $(8, 9)$. So, $x+y=8+9=17$. And that is our answer.

