Questions: 5/(y-7) - y/(y^2-9) - 2/(y-3)

Solution Steps

To combine the given fractions into a single fraction, we need to find a common denominator. The denominators are \(y-7\), \(y^2-9\), and \(y-3\). Notice that \(y^2-9\) can be factored as \((y-3)(y+3)\). The least common denominator (LCD) will be \((y-7)(y-3)(y+3)\). Rewrite each fraction with this common denominator and then combine them. Finally, simplify the resulting expression if possible.

We start with the expression: \[ \frac{5}{y-7} - \frac{y}{y^2-9} - \frac{2}{y-3} \] The denominators are \(y-7\), \(y^2-9\), and \(y-3\). Notably, \(y^2-9\) can be factored as \((y-3)(y+3)\).

The least common denominator (LCD) for the fractions is: \[ (y - 7)(y - 3)(y + 3) \]

We rewrite each fraction with the common denominator: \[ \frac{5}{y-7} = \frac{5(y-3)(y+3)}{(y-7)(y-3)(y+3)} \] \[ \frac{y}{y^2-9} = \frac{y}{(y-3)(y+3)} = \frac{y(y-7)}{(y-7)(y-3)(y+3)} \] \[ \frac{2}{y-3} = \frac{2(y-7)(y+3)}{(y-7)(y-3)(y+3)} \]

Combining the fractions gives: \[ \frac{5(y-3)(y+3) - y(y-7) - 2(y-7)(y+3)}{(y-7)(y-3)(y+3)} \]

After performing the necessary algebraic operations, we find the simplified numerator: \[ 2y^2 + 15y - 3 \] Thus, the combined expression simplifies to: \[ \frac{2y^2 + 15y - 3}{(y-7)(y-3)(y+3)} \]

Final Answer