Problem 68 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$7-\frac{4 y}{y+5}$$

Step-by-Step Solution

Answer

The simplified form of \(7 - \frac{4y}{y+5}\) is \(\frac{3y+35}{y+5}\).

Step 1: Identify the problem

The problem is \(7 - \frac{4y}{y+5}\). The issue here is that we have a whole number 7 which is being subtracted by a fraction \(\frac{4y}{y+5}\). As it stands, we can't just subtract \(7 - \frac{4y}{y+5}\) because they don't have the same denominators.

Step 2: Find a common denominator

To subtract or add fractions, they must have the same denominator (common denominator). In this case, to make the whole number 7 have the same denominator as \(\frac{4y}{y+5}\), we simply multiply it by \(\frac{y+5}{y+5}\). This won't change the value of the 7 because \(\frac{y+5}{y+5} = 1\). So, the problem now becomes \(\frac{7(y+5)}{y+5} - \frac{4y}{y+5}\)

Step 3: Perform the subtraction

Now that both components have the same denominator, they can be subtracted. The operation, therefore, simplifies to: \(\frac{7y+35}{y+5} - \frac{4y}{y+5}\) which further simplifies to \(\frac{7y+35-4y}{y+5}\) on the top line of the fraction.

Step 4: Simplify the fraction

Simplify the numerator (top line of the fraction) by combining similar terms. The fraction becomes \(\frac{3y+35}{y+5}\). The fraction can't be simplified any further since 3y and 35 don't share any common factors, and neither do 3y+35 and y+5. Thus the final simplified form is \(\frac{3y+35}{y+5}\).