Answer
$\dfrac{3(4y - 21)}{y(y - 7)}$
Restrictions: $x \ne 0, 7$
Work Step by Step
Factor all expressions in the original exercise:
$\frac{5y}{y(y - 7)} - \frac{4}{2(y - 7)} + \frac{9}{y}$
Before performing the addition, find the least common denominator (LCD) of the two fractions. The LCD is $2(y)(y-7)$.
Convert the original fractions to equivalent fractions using the LCD:
$=\dfrac{5y(2)}{y(y - 7)(2)} - \dfrac{4y}{2(y - 7)(y)} + \dfrac{9(2)(y - 7)}{y(2)(y - 7)}$
Multiply to simplify:
$=\dfrac{10y}{2y(y - 7)} - \dfrac{4y}{2y(y - 7)} + \dfrac{18y - 126}{2y(y - 7)}$
Add the numerators and retain the denominator:
$=\dfrac{10y - 4y + 18y - 126}{2y(y - 7)}$
Combine like terms:
$=\dfrac{24y - 126}{2y(y - 7)}$
Factor the numerator:
$=\dfrac{6(4y - 21)}{2y(y - 7)}$
Cancel out common factors in the numerator and denominator:
$=\dfrac{3(4y - 21)}{y(y - 7)}$
Restrictions on $x$ occur when the value of $x$ makes the fraction undefined, which means that the denominator becomes $0$.
Set the factors in the denominators equal to $0$ to find restrictions:
First factor:
$y = 0$
Second factor:
$y - 7 = 0$
Add $7$ to each side of the equation:
$y = 7$
Restrictions: $x \ne 0, 7$
