Question 952·Medium·Math

Choose $m=6$. Its denominator is $m-3=3$, so this value is allowed. Type $m=6$ and then $5m/(m-3)$ in Desmos. It prints $10$, so the original equation pairs $m=6$ with $k=10$. This $k$ also meets the given condition $k\ne5$.

Set $k=10$ and type the four proposed expressions for $m$ on separate Desmos lines. In the order shown, they print $3.5$, $7.142857\ldots$, $2$, and $6$. Only $m=\frac{3k}{k-5}$ returns the original $m=6$. Each other formula fails at this allowed pair, so it cannot be the answer. Choice D.

The target $m$ is in the denominator, so multiply both sides by the whole $m-3$. The given $m\ne3$ makes that multiplication valid:

Multiply $k$ by both terms inside the parentheses:

Subtract $5m$ from both sides so the terms containing the target $m$ can be gathered:

Add $3k$ to both sides, leaving both $m$-terms on the left:

Both $km$ and $-5m$ contain $m$. Pull it out as a common factor, so $m$ appears only once: $m(k-5)=3k$. Don’t divide by $m$ to remove it from the left: $m$ is the letter you need to isolate.

The given $k\ne5$ means $k-5$ is not zero. Divide both sides by $k-5$: $m=\frac{3k}{k-5}$. This expresses $m$ in terms of $k$ for every allowed value. Choice D.