Answer
$\frac{5y}{5y-12}$.
Work Step by Step
The given expression is
$=\frac{\frac{5y}{y^2-5y+6}}{\frac{3}{y-3}+\frac{2}{y-2}}$
Factor $y^2-5y+6$.
Rewrite the middle term $-5y$ as $-3y-2y$
$=y^2-3y-2y+6$
$=(y^2-3y)+(-2y+6)$
$=y(y-3)-2(y-3)$
$=(y-3)(y-2)$ plug into the given expression.
$=\frac{\frac{5y}{(y-3)(y-2)}}{\frac{3}{y-3}+\frac{2}{y-2}}$
Multiply the numerator and the denominator by $(y-3)(y-2)$.
$=\frac{(y-3)(y-2)}{(y-3)(y-2)}\cdot \frac{\frac{5y}{(y-3)(y-2)}}{\frac{3}{y-3}+\frac{2}{y-2}}$
Use the distributive property.
$=\frac{(y-3)(y-2)\cdot\frac{5y}{(y-3)(y-2)}}{(y-3)(y-2)\cdot\frac{3}{y-3}+(y-3)(y-2)\cdot\frac{2}{y-2}}$
Simplify.
$=\frac{5y}{3(y-2)+2(y-3)}$
$=\frac{5y}{3y-6+2y-6}$
$=\frac{5y}{5y-12}$.
