Question 208·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

Type $x^2+y^2=10$ and $y=x-1$ on separate lines. A solution must lie on both graphs, so click their intersections. Desmos shows approximately $(-1.679,-2.679)$ and $(2.679,1.679)$. In each ordered pair, the first number is $x$ and the second is $y$.

The positive intersection has $x\approx2.679$. Type $\frac{1+\sqrt{19}}{2}$; Desmos prints about $2.679$, so this expression matches its first coordinate. Don't stop at $x$: two choices use that same first coordinate but give different values for $y$.

The line says $y$ is one less than $x$. Replace $x$ with the matched expression:

Subtract $1$:

Type that expression. Desmos prints about $1.679$, matching the plotted second coordinate. So the solution is $\left(\frac{1+\sqrt{19}}{2},\frac{-1+\sqrt{19}}{2}\right)$. Choice D.