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Chapter 6: Problem 48

Perform the indicated operations. Simplify when possible $$ 3+\frac{y+2}{y-5} $$

Short Answer

Expert verified
\( \frac{4y - 13}{y - 5} \)

Step by step solution

01

Identify the expression

The given expression is \[ 3 + \frac{y+2}{y-5} \]
02

Find a common denominator

Since we have a fraction, to combine the terms, transform the integer 3 into a fraction with the same denominator as \(\frac{y+2}{y-5}\). Rewrite 3 as \( \frac{3(y-5)}{y-5} \), leading to \[ 3 + \frac{y+2}{y-5} = \frac{3(y-5)}{y-5} + \frac{y+2}{y-5} \]
03

Combine the fractions

Add the fractions by combining the numerators over the common denominator: \[ \frac{3(y-5)}{y-5} + \frac{y+2}{y-5} = \frac{3y - 15 + y + 2}{y-5} \]
04

Simplify the numerator

Simplify the numerator by combining like terms: \[ \frac{3y - 15 + y + 2}{y-5} = \frac{4y - 13}{y-5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common denominator
To simplify rational expressions like \( 3 + \frac{y+2}{y-5} \), the first step is to find a common denominator for combining terms. The reason for this is to ensure all parts of the expression can be worked with uniformly.
When given an integer such as 3, we must convert it into a fraction that has the same denominator as the existing fraction.
In our problem, the existing fraction is \( \frac{y+2}{y-5} \). So, we rewrite 3 as \( \frac{3(y-5)}{y-5} \).
This step makes it possible to combine the two fractions, since they now share a common denominator. Finding a common denominator makes it possible to perform operations across fractions safely and effectively.
combining fractions
Combine fractions by ensuring they share a common denominator. Once fractions have the same denominator, you can merge them by adding or subtracting the numerators directly.
In our example, after rewriting 3 as \( \frac{3(y-5)}{y-5} \), we get: \[ 3 + \frac{y+2}{y-5} = \frac{3(y-5)}{y-5} + \frac{y+2}{y-5} \]
To combine these fractions, we add the numerators together while keeping the denominators the same:
\[ \frac{3(y-5)}{y-5} + \frac{y+2}{y-5} = \frac{3y - 15 + y + 2}{y-5} \].
This results in a single fraction. Combining fractions this way ensures consistency throughout the expression.
simplifying numerators
After combining fractions, the next step is to simplify the numerator. Simplification involves combining like terms to make the expression clearer and more manageable.
In our final combined fraction: \[ \frac{3y - 15 + y + 2}{y-5} \],
we will combine like terms:
\[3y + y - 15 + 2 = 4y - 13 \].
This finally gives us: \[ \frac{4y - 13}{y-5} \].
Simplifying the numerator is essential for reaching the simplest form of the expression, making it easier to understand and work with for further mathematical operations or analysis.

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