Problem 94 - Elementary Algebra

Question

For exercises 91-94, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Simplify: \(\frac{y+4}{2 y+10}-\frac{5}{y^{2}-25}\) Incorrect Answer: The least common denominator is \(2(y-5)(y+5)\). $$ \begin{aligned} &\frac{y+4}{2 y+10}-\frac{5}{y^{2}-25} \\ &=\frac{y+4}{2(y+5)} \cdot \frac{(y-5)}{(y-5)}-\frac{5}{(y-5)(y+5)} \cdot \frac{2}{2} \\ &=\frac{y^{2}+4 y-5 y-20}{2(y-5)(y+5)}-\frac{10}{2(y-5)(y+5)} \\ &=\frac{y^{2}-y-30}{2(y-5)(y+5)} \\ &=\frac{(y+6)(y-5)}{2(y-5)(y+5)} \\ &=\frac{y+6}{2(y+5)} \end{aligned} $$

Step-by-Step Solution

Answer

The correct simplified form is \( \frac{y-6}{2(y-5)} \).

Step 1: Identify the mistake

The least common denominator (LCD) is incorrectly identified as \(2(y-5)(y+5)\). The correct LCD should be \(2(y+5)(y-5)\). The terms involving the different factors need to be properly accounted for.

Step 2: Rewrite the denominators

The denominators need to be written properly:\[ \frac{y+4}{2y+10} - \frac{5}{y^2-25} \] Simplify them: \[ 2y + 10 = 2(y + 5) \] \[ y^2 - 25 = (y - 5)(y + 5) \] So, the expression becomes:\[ \frac{y+4}{2(y+5)} - \frac{5}{(y-5)(y+5)} \]

Step 3: Find the correct LCD

Identify the correct least common denominator for these fractions:\[ \frac{y+4}{2(y+5)} \text{ and } \frac{5}{(y-5)(y+5)} \] The correct least common denominator is:\[ 2(y+5)(y-5) \]

Step 4: Rewrite fractions with the LCD

Adjust each fraction to have the correct LCD: \[ \frac{y+4}{2(y+5)} = \frac{y+4}{2(y+5)} \frac{y-5}{y-5} = \frac{(y+4)(y-5)}{2(y+5)(y-5)} \] \[ \frac{5}{(y-5)(y+5)} = \frac{5}{(y-5)(y+5)} \frac{2}{2} = \frac{10}{2(y+5)(y-5)} \]

Step 5: Combine the fractions

Combine the fractions over the common denominator: \[ \frac{(y+4)(y-5)}{2(y+5)(y-5)} - \frac{10}{2(y+5)(y-5)} \] \[ \frac{(y+4)(y-5) - 10}{2(y+5)(y-5)} \]

Step 6: Simplify the numerator

Expand and simplify the numerator:\[ (y+4)(y-5) = y^2 - 5y + 4y - 20 = y^2 - y - 20 \] So the fraction becomes:\[ \frac{y^2 - y - 20 - 10}{2(y+5)(y-5)} \] \[ \frac{y^2 - y - 30}{2(y+5)(y-5)} \]

Step 7: Factor the numerator

Factor the numerator if possible:\[ y^2 - y - 30 \] \[ = (y + 5)(y - 6) \] So the expression now is:\[ \frac{(y+5)(y-6)}{2(y+5)(y-5)} \]

Step 8: Cancel out common factors

Cancel out common factors in the numerator and the denominator:\[ \frac{(y+5)(y-6)}{2(y+5)(y-5)} = \frac{y-6}{2(y-5)} \]

Key Concepts

Least Common Denominator

Understanding how to find the least common denominator (LCD) is crucial when working with algebraic fractions. The LCD is essential because it allows fractions to be combined or subtracted by giving them a common baseline.

To determine the LCD, look at the denominators of each fraction and identify their factors. For example:

• For the fraction \( \frac{y+4}{2y+10} \), the denominator can be factored as \( 2(y + 5) \).
• For the fraction \( \frac{5}{y^2 - 25} \), the denominator can be factored as \( (y-5)(y+5) \), which is a difference of squares.

Once the denominators are factored, the LCD is the product of all unique factors at their highest powers. In our example, the LCD for \( 2(y + 5) \) and \( (y - 5)(y + 5) \) is \( 2(y+5)(y-5)\). Using this common denominator allows us to rewrite the fractions in a form that can be easily subtracted or added together.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This often includes combining like terms, factoring, and reducing fractions. For algebraic fractions, simplification often involves common denominators and reducing the fraction.

In the given problem, we rewrite each term with a common denominator before simplification:

• Rewrite \( \frac{y+4}{2(y+5)} \cdot \frac{(y-5)}{(y-5)} = \frac{(y+4)(y-5)}{2(y+5)(y-5)} \)
• Rewrite \( \frac{5}{(y-5)(y+5)} \cdot \frac{2}{2} = \frac{10}{2(y+5)(y-5)} \).

We then subtract these fractions and expand the numerator:
\( \frac{(y+4)(y-5) - 10}{2(y+5)(y-5)} = \frac{y^2-y-20 - 10}{2(y+5)(y-5)} = \frac{y^2-y-30}{2(y+5)(y-5)} \).
We factor the numerator and cancel common terms to fully simplify:
\( \frac{(y+6)(y-5)}{2(y+5)(y-5)} = \frac{y+6}{2(y+5)} \).

This process makes it clear that achieving the simplest form involves consistent use of algebraic rules.

Factoring Algebraic Expressions

Factoring is the process of breaking down an expression into simpler expressions that multiply together to give the original. It's a fundamental skill in algebra.

In our example, here are the key steps to factoring used:

• Factoring out common terms, like in \(2y + 10 = 2(y + 5) \).
• Recognizing patterns such as the difference of squares, with \( y^2 - 25 = (y - 5)(y + 5) \).
• Breaking down quadratic expressions, like \( y^2 - y - 30 = (y + 5)(y - 6) \).

Factoring allows us to simplify expressions and solve equations more efficiently. Applying these techniques:

• We can adjust and rewrite complex fractions
• Prepare expressions for combination and simplification
• Identify and cancel out common factors, which helps in reducing expressions to their simplest form.

This process highlights the importance of recognizing factoring opportunities in algebraic problems.