Question 202·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

Type $y=x$. Desmos draws a diagonal line. Every point on it has equal $x$- and $y$-coordinates, so a solution to this system must look like $(x,x)$.

Add $(x-1)^2+(y-8)^2=37$ as written. Desmos draws a circle crossing the line twice. Click both intersections to see $(2,2)$ and $(7,7)$. Each intersection solves both equations, but you must compare the coordinate the question asks for. The first coordinates are $2$ and $7$, so the greatest possible value of $x$ is $7$. Choice C.

The line says $y=x$, so use substitution: replace $y$ with $x$ in the circle equation. This keeps only points that can lie on both graphs:

Expand both squares, keeping their negative middle terms:

Combine like terms, which are terms with the same power of $x$:

Subtract $37$ from both sides:

Divide every term by $2$:

To factor $x^2-9x+14$, find two numbers that multiply to $14$ and add to $-9$. They are $-7$ and $-2$, so:

By the zero-product rule, if a product is $0$, at least one factor is $0$. Set each factor to $0$:

Solve each equation:

The greater possible value of $x$ is $7$. Choice C.