### 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 is a problem involving quadratic equations and circles.
I can rewrite the equation as \[x^2 - x + y^2 - y = 0\]and try to complete the square for both $x$ and $y$.
To complete the square for $x$, I need to add $(1/2)^2 = 1/4$ to both sides of the equation.
To complete the square for $y$, I need to add $(1/2)^2 = 1/4$ to both sides of the equation as well.
So the equation becomes \[(x - 1/2)^2 + (y - 1/2)^2 = 1/2\]
Now I recognize this as the equation of a circle with center $(1/2, 1/2)$ and radius $\sqrt{1/2}$.
The largest value of $x$ for which the equation has a solution is when $x$ is at the rightmost point of the circle.
That means $x - 1/2 = \sqrt{1/2}$, or $x = \sqrt{1/2} + 1/2$.