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Chapter 8: Problem 246

In the following exercises, simplify. $$ \frac{3 y-1}{y+4}-2 $$

Short Answer

Expert verified
The simplified expression is \( \frac{y-9}{y+4} \).

Step by step solution

01

Identify the Expression

Start by examining the given expression: \( \frac{3 y-1}{y+4} - 2 \). Recognize that it has two parts: a fraction and a constant.
02

Create a Common Denominator

To combine the terms, we need a common denominator. Here, the common denominator is \( y+4 \). Rewrite the constant so it has the same denominator: $$ -2 = \frac{-2(y+4)}{y+4} = \frac{-2y-8}{y+4} $$.
03

Combine the Fractions

Now that both terms have the same denominator, combine the fractions: $$ \frac{3y-1}{y+4} - \frac{2y+8}{y+4} = \frac{(3y-1) - (2y+8)}{y+4} = \frac{3y-1-2y-8}{y+4} $$.
04

Simplify the Numerator

Simplify the numerator by combining like terms: $$ 3y-2y-1-8 = y-9 $$. So the expression becomes: $$ \frac{y-9}{y+4} $$.

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Key Concepts

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

Fraction Subtraction
When subtracting fractions, it's essential to handle each part of the expression with care. Fractions are different from integers or simple algebraic terms because they require a common denominator. Let's look at a fraction subtraction example: \ The expression given is: \ $$ \frac{3y-1}{y+4} - 2 $$ \ To subtract, we first need to transform the constant term into a fraction. \ This is done by expressing it with the same denominator as the given fraction.
Common Denominator
To combine fractions, we need a common denominator. The common denominator helps streamline the operations by aligning the terms under a single denominator. Let's rewrite the constant term in our exercise helpfully as a fraction: \ The given fraction: \ $$ \frac{3y-1}{y+4} $$ \ We need to express \ $$ -2 $$ \ with a common denominator: \ $$ -2 = \frac{-2(y+4)}{y+4} = \frac{-2y-8}{y+4} $$ \ Now, both terms have the denominator \ $$ y+4 $$.
Combining Like Terms
After establishing a common denominator, we can combine fractions by subtracting the numerators. Combining like terms involves simplifying expressions by grouping similar terms together. Consider our example: \ Combine the fractions: \ $$ \frac{3y-1}{y+4} - \frac{2y+8}{y+4} $$ \ Subtract numerators directly: \ $$ \frac{(3y-1) - (2y+8)}{y+4} = \frac{3y-1-2y-8}{y+4} $$.
Simplifying Algebraic Expressions
The final step involves simplifying the algebraic expression by combining like terms in the numerator. Let鈥檚 break this down: \ We had: \ $$ \frac{3y-1-2y-8}{y+4} $$ \ Firstly, combine the 'y' terms: \ $$ 3y - 2y = y $$ \ Then, combine the constants: \ $$ -1 - 8 = -9 $$ \ Our simplified numerator becomes: \ $$ y - 9 $$ \ Resulting in the final expression: \ $$ \frac{y-9}{y+4} $$ \ This completes the process of simplifying the given algebraic expression by following through fraction subtraction, establishing a common denominator, combining like terms, and finally simplifying.

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Most popular questions from this chapter

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