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Chapter 7: Problem 68

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

Short Answer

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

Step by step solution

01

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.
02

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 \(\frac{7(y+5)}{y+5} - \frac{4y}{y+5}\)
03

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.
04

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 \(\frac{3y+35}{y+5}\).

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Most popular questions from this chapter

Describe two similarities between the following problems: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x+6}-1$$

Solve: \(x^{2}-12 x+36=0 .\) (Section 6.6, Example 4)

Factor completely: \(81 x^{4}-1\)

Explain how to subtract rational expressions when denominators are the same. Give an example with your explanation.
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