Question 15·Medium·Advanced Math

Choose $x=2$ and $y=1$. They're allowed because $2(2)\ne3(1)$. Type $x_1=2$, $y_1=1$, and the original fraction into Desmos, using $x_1$ and $y_1$ in place of $x$ and $y$. Desmos prints $7$. Save that as a check; one pair alone cannot prove equivalence for all allowed values.

The numerator is a difference of squares: let $A=2x$ and $B=3y$, so $4x^2=A^2$ and $9y^2=B^2$. Use $A^2-B^2=(A-B)(A+B)$; the middle products cancel when you multiply those factors back out. So one factor matches the denominator, since $A-B=2x-3y$:

The whole factor $A-B=2x-3y$ matches the denominator. The given $2x\ne3y$ means that factor isn't zero, so you can divide the top and bottom by it:

Type $2x_1+3y_1$ below the original in Desmos. It prints $7$, matching your check. The factoring and cancellation, not the single test, show that the equivalent expression is $2x+3y$ for every allowed pair. Choice D.