Simplify 3/(y-5)-(5y-16)/(y^2-7y+10)

Consider the form ${\displaystyle x^2+bx+c}$. Find a pair of integers whose product is ${\displaystyle c}$ and whose sum is ${\displaystyle b}$. In this case, whose product is ${\displaystyle 10}$ and whose sum is ${\displaystyle -7}$.

Write the factored form using these integers.

Multiply ${\displaystyle \frac{3}{y-5}}$ by ${\displaystyle \frac{y-2}{y-2}}$.

Combine the numerators over the common denominator.

Apply the distributive property.

Multiply ${\displaystyle 3}$ by ${\displaystyle -2}$.

Apply the distributive property.

Multiply ${\displaystyle 5}$ by ${\displaystyle -1}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle -16}$.

Subtract ${\displaystyle 5y}$ from ${\displaystyle 3y}$.

Add ${\displaystyle -6}$ and ${\displaystyle 16}$.

Factor ${\displaystyle 2}$ out of ${\displaystyle -2y+10}$.

Factor ${\displaystyle -1}$ out of ${\displaystyle -y}$.

Rewrite ${\displaystyle 5}$ as ${\displaystyle -1(-5)}$.

Factor ${\displaystyle -1}$ out of ${\displaystyle -\left(y\right)-1(-5)}$.

Rewrite ${\displaystyle -(y-5)}$ as ${\displaystyle -1(y-5)}$.

Cancel the common factor.

Rewrite the expression.