### 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: I notice that the equation is symmetric in $x$ and $y$, so I can swap them without changing the solutions.
This means that the largest value of $x$ is the same as the largest value of $y$, so I only need to find one of them.
I also notice that the equation is a circle with center $\left(\frac{1}{2}, \frac{1}{2}\right)$ and radius $\frac{\sqrt{2}}{2}$, since I can rewrite it as
\[\left(x - \frac{1}{2}\right)^2 + \left(y - \frac{1}{2}\right)^2 = \frac{1}{2}\]
To find the largest value of $x$, I need to find the point on the circle that is farthest to the right, which is when the circle touches the vertical line $x = \frac{1}{2} + \frac{\sqrt{2}}{2}$.
Therefore, the largest value of $x$ is $\frac{1}{2} + \frac{\sqrt{2}}{2}$.