Problem 79 - Introductory Algebra for College Students

Question

Simplify each rational expression. $$\frac{9-y^{2}}{y^{2}-3(2 y-3)}$$

Step-by-Step Solution

Answer

The simplified rational expression is \( \frac{-(y+3)}{(y-3)} \).

Step 1: Rewrite the expression

Rewrite the expression \( \frac{9-y^{2}}{y^{2}-3(2 y-3)} \) as \( \frac{9-y^{2}}{y^{2}-6y+9} \) by distributing the -3 in the denominator.

Step 2: Rearrange the numerator

Refactor \(9 - y^{2}\) in the denominator and it becomes \(-(y^{2} - 9)\).

Step 3: Factor the expressions

Apply the difference of squares to both the numerator and denominator. The rational expression is then: \( \frac{-(y+3)(y-3)}{(y-3)(y-3)} \).

Step 4: Cancel common terms

Cancel the common term in the numerator and the denominator (y-3). The simplified rational expression is: \( \frac{-(y+3)}{(y-3)} \).