Question

subtract the following fractions. simplify your answer as much as possible.
\\(\frac{12y}{y^2 - 36} - \frac{6}{y - 6}\\)

Explanation:

Step1: Factor the denominator

First, factor the denominator \(y^2 - 36\) using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\). So, \(y^2 - 36=(y + 6)(y - 6)\). The expression becomes \(\frac{12y}{(y + 6)(y - 6)}-\frac{6}{y - 6}\).

Step2: Find a common denominator

The common denominator of \((y + 6)(y - 6)\) and \(y - 6\) is \((y + 6)(y - 6)\). Rewrite the second fraction with the common denominator: \(\frac{6}{y - 6}=\frac{6(y + 6)}{(y + 6)(y - 6)}\).

Step3: Subtract the fractions

Now subtract the two fractions: \(\frac{12y}{(y + 6)(y - 6)}-\frac{6(y + 6)}{(y + 6)(y - 6)}=\frac{12y-6(y + 6)}{(y + 6)(y - 6)}\).

Step4: Simplify the numerator

Simplify the numerator: \(12y-6(y + 6)=12y-6y - 36 = 6y-36\). Factor out 6 from the numerator: \(6y - 36 = 6(y - 6)\).

Step5: Cancel common factors

The expression becomes \(\frac{6(y - 6)}{(y + 6)(y - 6)}\). Cancel out the common factor \((y - 6)\) (assuming \(y
eq6\)): \(\frac{6}{y + 6}\).

Answer:

\(\frac{6}{y + 6}\)