Question 597·Hard·Algebra

No Desmos needed. Keep $u$ as a letter and use the slope formula. In $q(a)=b$, $a$ is the input and $b$ is the output, so the given values give the input-output points $(u-4,7u+2)$ and $(3u+1,-5u+16)$.

In $q(x)=rx+s$, $r$ is the slope: output change divided by input change. Use the second point minus the first on both top and bottom:

Each minus sign before parentheses changes every sign inside. In particular, subtracting $u-4$ adds $4$:

Combine the $u$ terms and the number terms on each side of the fraction bar:

So the coefficient $r$ is $\frac{-12u+14}{2u+5}$. Choice D.

The correct expression must work for an allowed value of $u$. Try $u=1$; the two inputs will be different. Type $u=1$, then the four expressions from the given function values. Desmos shows $-3,9,4,11$ in that order, giving the points $(-3,9)$ and $(4,11)$.

Type the output change over the input change, keeping the points in the same order. Desmos shows $0.285714\ldots$; its fraction button gives $\frac27$. That is $r$ when $u=1$, not yet a formula for $r$.

Type the four choices with $u=1$ still set. In listed order, Desmos shows $-\frac27$, $-\frac23$, $-\frac{26}{3}$, and $\frac27$ using the fraction buttons. Only one matches the slope you found. Its expression is $r=\frac{-12u+14}{2u+5}$. Choice D.