### Instruction: Find the largest value of $x$ for which
\[x^2 + y^2 = x + y\]has a solution, if $x$ and $y$ are real.### Response: This problem looks like it involves circles, since the equation has $x^2 + y^2$ on one side.
I wonder if I can rewrite the equation in a more familiar form, like $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ is the center and $r$ is the radius of the circle.
To do that, I need to complete the square for both $x$ and $y$ terms.
I notice that the equation has $x$ and $y$ on the right side, so I can move them to the left side and get \[x^2 - x + y^2 - y = 0\]
Now I can group the $x$ and $y$ terms and add some constants to both sides to complete the square.
For the $x$ terms, I need to add $(1/2)^2 = 1/4$ to both sides, since $(x-1/2)^2 = x^2 - x + 1/4$.
For the $y$ terms, I need to add $(1/2)^2 = 1/4$ to both sides, since $(y-1/2)^2 = y^2 - y + 1/4$.
So the equation becomes \[(x-1/2)^2 + (y-1/2)^2 = 1/2\]
Now I see that this is a circle with center $(1/2, 1/2)$ and radius $\sqrt{1/2}$.
I want to find the largest value of $x$ that satisfies this equation, so I need to find the rightmost point on the circle.
That point will have the same $y$-coordinate as the center, which is $1/2$, and the $x$-coordinate will be the radius more than the center, which is $1/2 + \sqrt{1/2}$.
Therefore, the largest value of $x$ is $1/2 + \sqrt{1/2}$.