Problem 47 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified result of the given expression is \( \frac{8y+16}{y(y+4)} \).

Step 1: Find the common denominator and adjust the numerators

The common denominator is the product of \(y\) and \(y+4\). For the first fraction, multiply the numerator and the denominator by \(y+4\). For the second fraction, do the same but with \(y\). It results in: \(\frac{(y+4)^2}{y(y+4)}-\frac{y^2}{y(y+4)}\).

Step 2: Subtract the fractions

Now that both fractions have the same denominator, you can subtract them. Write down the common denominator once and subtract the numerators: \(\frac{(y+4)^2 - y^2}{y(y+4)}\).

Step 3: Simplify the result

Expand \((y+4)^2\) to get \(y^2+8y+16\). Then subtract \(y^2\) from it to get the new numerator: \(8y+16\). The final result is: \(\frac{8y+16}{y(y+4)}\).