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Chapter 8: Problem 213

In the following exercises, subtract. $$ \frac{y-4}{y+1}-\frac{1}{y+7} $$

Short Answer

Expert verified
\( \frac{y^2 + 2y - 29}{(y+1)(y+7)} \)

Step by step solution

01

Find a common denominator

To subtract the fractions, find a common denominator. In this case, the common denominator for \(y+1\) and \(y+7\) is \( (y+1)(y+7) \).
02

Rewrite each fraction with the common denominator

Rewrite each fraction with the common denominator. This gives: \[ \frac{(y-4)(y+7)}{(y+1)(y+7)} - \frac{1(y+1)}{(y+1)(y+7)} \]
03

Distribute and simplify the numerators

Expand the numerators by distributing: \[ \frac{y^2 + 7y - 4y - 28}{(y+1)(y+7)} - \frac{y+1}{(y+1)(y+7)} \] This simplifies to \[ \frac{y^2 + 3y - 28}{(y+1)(y+7)} - \frac{y+1}{(y+1)(y+7)} \]
04

Combine the fractions

Combine the numerators over the common denominator: \[ \frac{y^2 + 3y - 28 - (y + 1)}{(y+1)(y+7)} \] Simplify the numerator: \[ y^2 + 3y - 28 - y - 1 = y^2 + 2y - 29 \]
05

Write the final answer

Combine the simplified numerator and the common denominator to get the final answer: \[ \frac{y^2 + 2y - 29}{(y+1)(y+7)} \]

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Key Concepts

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

Common Denominator
When you subtract algebraic fractions, you need to have a common denominator. This makes the subtraction process easier.

Think of the common denominator as a common base for both fractions. In our example, the denominators are \(y+1\) and \(y+7\). We find the common denominator by multiplying these two terms together, giving us \( (y+1)(y+7) \).
  • First, the denominators \(y+1\) and \(y+7\) are taken individually.
  • Next, multiply them: \( (y+1)(y+7) \).
This common denominator allows both fractions to be written over the same baseline, simplifying the subtraction process.
Simplifying Fractions
After finding the common denominator, each fraction must be rewritten with this baseline.

For example, rewrite \( \frac{y-4}{y+1} \) and \( \frac{1}{y+7} \) to have the new common denominator \( (y+1)(y+7) \). This step allows us to combine the fractions:
\[ \frac{(y-4)(y+7)}{(y+1)(y+7)} \- \frac{(1)(y+1)}{(y+1)(y+7)} \] Expanding using distribution (covered next) gives:
\[ \frac{y^2 + 7y - 4y - 28}{(y+1)(y+7)} - \frac{y+1}{(y+1)(y+7)} \]
Once the denominators are the same, you can directly combine the numerators:
\[ \frac{y^2 + 3y - 28 - (y + 1)}{(y+1)(y+7)} \]
Finally, simplify the expression in the numerator to find the combined fraction.
Distributing in Algebra
Distribution is key when dealing with polynomial expressions. It involves multiplying each term inside a parenthesis by the term outside.

In our example, we distribute \(y-4 \) and \( y+7 \):
\[ (y-4)(y+7) = y^2 + 7y - 4y - 28 \] To distribute properly:
  • Multiply \( y \) by each term in \( y+7\).
  • Multiply \( -4 \) by each term in \( y+7 \) and combine like terms.
Distribution helps simplify the numerator, making it easier to find the final fraction: \[ y^2 + 2y - 29 \] after combining like terms. This final step ensures our algebraic expression is simplified and ready for subtraction.

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