Problem 41 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{5 y}{y^{2}-9}-\frac{4}{y+3}$$

Step-by-Step Solution

Answer

The result is \(\frac{y + 12}{(y+3)(y-3)}\)

Step 1: Identify the denominators

The denominators of the fractions are \(y^{2}-9\) and \(y+3\), respectively. They are not the same, so we need to find a common denominator to make subtraction possible.

Step 2: Find common denominator

The expression \(y^{2}-9\) can be factorized to \((y+3)(y-3)\), since it's a difference of two squares. Therefore, the common denominator of the two fractions is \((y+3)(y-3)\).

Step 3: Transform the fractions

To make the denominators the same we multiply the second fraction by \((y-3)/(y-3)\) which gives:\[\frac{5 y}{(y+3)(y-3)}- \frac{4(y-3)}{(y+3)(y-3)}\]

Step 4: Subtract the fractions

Now that the denominators are the same, subtract the fractions:\[\frac{5 y - 4(y-3)}{(y+3)(y-3)}\]

Step 5: Simplify the result

Simplify the numerator of the fraction:\[\frac{5 y - 4y + 12}{(y+3)(y-3)}=\frac{y + 12}{(y+3)(y-3)}\]