Once you write $r(u-4)+s=7u+2$ and $r(3u+1)+s=-5u+16$, think of them as a system of equations in the unknowns $r$ and $s$.

Notice that both equations have a $+s$ term. What operation can you do with the two equations so that the $s$ terms cancel out?

After eliminating $s$, you should have an equation of the form $r(\text{expression in }u)=\text{another expression in }u$. Solve this equation for $r$ by dividing.

In Desmos, on one line type (-5u+16)-(7u+2) and note the simplified expression it shows. On another line type (3u+1)-(u-4) and note that simplified expression.

On a new line, type (( -5u+16 ) - ( 7u+2 )) / ( (3u+1) - (u-4) ). Desmos will simplify this fraction to an expression in terms of $u$; that expression is $r$. Compare what Desmos shows to the four answer choices and select the matching one.

We know $q(x)=rx+s$.

• For $x=u-4$:
  • $q(u-4)=r(u-4)+s$
  • But the problem says $q(u-4)=7u+2$.
  • So:

• For $x=3u+1$:
  • $q(3u+1)=r(3u+1)+s$
  • But the problem says $q(3u+1)=-5u+16$.
  • So:

Now you have a system of two equations in the unknowns $r$ and $s$.

Subtract the first equation from the second to eliminate $s$:

On the left side, $s-s=0$, so only the $r$ terms remain:

Factor $r$ on the left:

First simplify inside the brackets on the left:

So the left side becomes $r(2u+5)$.

Now simplify the right side:

So the equation is now:

To isolate $r$, divide both sides by $2u+5$ (assuming $2u+5\ne 0$):

This matches choice D.