Problem 42 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{8 y}{y^{2}-16}-\frac{5}{y+4}$$

Step-by-Step Solution

Answer

The simplified expression from the given exercise is \(\frac{3y+20}{(y-4)(y+4)}\)

Step 1: Identify the Difference of Squares

The denominator of the first fraction, \(y^{2}-16\), is a difference of perfect squares, which can be factored as \((y-4)(y+4)\).

Step 2: Rewrite the first fraction

Rewrite the first fraction using the new denominator. The fraction becomes: \[\frac{8y}{(y-4)(y+4)}\].

Step 3: Find a common denominator

To subtract fractions, a common denominator is needed. In this case, the common denominator should be \((y-4)(y+4)\), which is achieved by writing the second fraction as \[\frac{5(y-4)}{(y-4)(y+4)}]\].

Step 4: Subtract the fractions

Subtract the fractions by subtracting the numerators while keeping the common denominator as is: \[\frac{8y-5(y-4)}{(y-4)(y+4)}\].

Step 5: Simplify the expression

Simplify the expression by distributing the -5 in the numerator: \[\frac{8y-5y+20}{(y-4)(y+4)}=\frac{3y+20}{(y-4)(y+4)}\]