Question 979·Medium·Math

Choose $q=6$. Type $q=6$ and $3/q$; Desmos shows $0.5$. That leaves a positive $0.5$ for $2/p$, as required because $p$ is positive.

Keep $q=6$ and type $\frac{2}{p_1}+\frac{3}{q}\sim1\{p_1>0\}$. The $\sim$ asks Desmos to fit $p_1$, and the braces require a positive result. Under PARAMETERS, Desmos shows $p_1=4$. So the original equation gives $p=4$ when $q=6$.

Type the four answer expressions as a list, in their given order. Desmos shows approximately $[1.3333,-0.1176,0.25,4]$. Only the fourth prediction matches the $p=4$ found from the original equation. Test every formula at the same valid input. So $p=\dfrac{2q}{q-3}$. Choice D.

Treat $q$ as fixed. To leave the fraction containing $p$ alone, subtract $3/q$ from both sides:

Since $q$ is positive, $q/q=1$. Rewrite $1$ with denominator $q$:

The fractions have the same denominator, so subtract their numerators and keep $q$ underneath:

Because $p$ and $q$ are positive, neither denominator is zero. Multiply both sides by $pq$:

Here $2q>0$ and $p>0$, so $q-3$ must be positive. Divide by $q-3$:

This expresses $p$ in terms of $q$ for the positive numbers in the question. Choice D.