If $(x,y)$ and $(y,x)$ must represent the same ordered pair, what relationship must hold between $x$ and $y$?

Substitute the relationship between $x$ and $y$ into $x^2+y^2=50$, then compare the possible values with the answer choices.

Enter x^2+y^2=50 and x+y=p in Desmos. Desmos will create a slider for $p$.

Adjust $p$ until the line touches the circle at exactly one point. One such slider value is positive, and another is negative.

Click the single intersection point and read its $x$-coordinate. Compare the displayed coordinate with the answer choices.

Both equations are unchanged when $x$ and $y$ are switched. Therefore, if $(x,y)$ is a solution, then $(y,x)$ is also a solution.

Because the system has exactly one ordered-pair solution, these two ordered pairs must be the same. Thus,

Replace $y$ with $x$ in $x^2+y^2=50$:

Therefore,

For $x=y=5$, the linear equation requires

When $p=10$, substituting $y=10-x$ into the first equation gives

which simplifies to

So $x=5$ gives exactly one solution when $p=10$. Among the choices, $5$ could be the value of $x$.