### 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 it has $x^2 + y^2$ on one side.
I wonder if I can rewrite the equation in a way that makes the circle more clear.
Maybe I can complete the square for both $x$ and $y$.
To do that, I need to move the linear terms to the other side, so I get \[x^2 - x + y^2 - y = 0\]
Now I can add half of the coefficients of $x$ and $y$ squared to both sides, so I get \[(x^2 - x + 1/4) + (y^2 - y + 1/4) = 1/2\]
This simplifies to \[(x - 1/2)^2 + (y - 1/2)^2 = 1/2\]
Now I see that this is the equation of a circle with center $(1/2, 1/2)$ and radius $\sqrt{1/2}$.
The question is asking for the largest value of $x$ that satisfies this equation, which means the rightmost point on the circle.
I can find that by plugging in $y = 1/2$, since that is the vertical coordinate of the center.
That gives me \[(x - 1/2)^2 + (1/2 - 1/2)^2 = 1/2\] which simplifies to \[(x - 1/2)^2 = 1/2\]
Taking the square root of both sides, I get \[x - 1/2 = \pm \sqrt{1/2}\]
Adding 1/2 to both sides, I get \[x = 1/2 \pm \sqrt{1/2}\]
The largest value of $x$ is the one with the positive sign, so the final answer is \[x = 1/2 + \sqrt{1/2}\]