Question 6·Medium·Advanced Math

Try $x=1$ and $y=1$. Type them as $x_1$ and $y_1$ so Desmos uses them as numbers rather than graph variables. Then type $2x_1+3y_1$. Desmos shows $5$, so the denominator is not zero and this pair is allowed.

Type the original fraction using $x_1$ and $y_1$. Desmos shows $-1$. Any equivalent expression must give $-1$ at this allowed pair.

The first option is already on screen. Type the other three using $x_1$ and $y_1$. Desmos shows $5$, $0.5$, $3.5$, and $-1$, in that order. The first three fail at an allowed pair, so they cannot work for all allowed values. The only match is $2x-3y$. Choice D.

The numerator is a difference of squares: $4x^2$ is $(2x)^2$, and $9y^2$ is $(3y)^2$. Write those squares explicitly:

A difference of squares follows $A^2-B^2=(A-B)(A+B)$. Here $A=2x$ and $B=3y$, so the factors have opposite signs:

Replace the numerator with its factored form. The denominator now appears as a whole factor on top:

Because the original expression is defined only when $2x+3y\ne0$, divide the top and bottom by that factor:

Cancel whole factors, not pieces of sums. The equivalent expression is $2x-3y$ for every allowed pair. Choice D.