Problem 47 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{9 y}{x-4}+y $$

Step-by-Step Solution

Answer

\( \frac{yx + 5y}{x-4} \)

Step 1: Identify Common Denominator

To combine the terms \( \frac{9y}{x-4} \) and \( y \), we need a common denominator. The first term already has a denominator of \( x-4 \), so we will write \( y \) as \( \frac{y(x-4)}{x-4} \) to give it the same denominator.

Step 2: Rewrite the Second Term

Rewrite \( y \) as \( \frac{y(x-4)}{x-4} \). This gives us two fractions with the same denominator: \( \frac{9y}{x-4} + \frac{y(x-4)}{x-4} \).

Step 3: Add the Fractions

Combine the numerators of the fractions. This results in: \( \frac{9y + y(x-4)}{x-4} \).

Step 4: Simplify the Numerator

Distribute \( y \) in the second numerator term: \( y(x-4) = yx - 4y \). The numerator becomes \( 9y + yx - 4y \).

Step 5: Combine Like Terms

Combine like terms in the numerator. \( 9y - 4y = 5y \), so the expression becomes \( yx + 5y \). The combined expression is now \( \frac{yx + 5y}{x-4} \).

Step 6: Final Simplified Expression

There are no common factors to factor out further, so the simplest form of the expression is \( \frac{yx + 5y}{x-4} \).

Key Concepts

Common Denominators

Finding a common denominator is crucial when dealing with fraction addition. A common denominator allows us to combine fractions by aligning their bases, facilitating easier arithmetic operations. Think of it as translating numbers into the same language so they can be easily compared and combined.

• When adding fractions, check what’s beneath each fraction line – that's your denominators. This exercise started with \[ \frac{9y}{x-4} + y \]where the challenge was the single denominator under only one fraction.
• To combine these into a single expression, both parts must share a common denominator, like learning a new language to speak with others.

By transforming the whole term \(y\) into \( \frac{y(x-4)}{x-4} \), we align the parts under one common denominator, \(x-4\). This makes adding them straight forward. Once they share the same ‘line of communication,’ we can press forward with other computations.

Fraction Addition

Adding fractions follows a logical pattern once their denominators align.We essentially stack the numerators over the common denominator. Let's explore how this works with our example:

• With fractions like \( \frac{9y}{x-4} \, \text{and} \, \frac{y(x-4)}{x-4} \), both have the denominator \(x-4\).
• To add, maintain the denominator and combine the numerators: \( 9y + y(x-4) \).

This method simplifies your life by keeping things organized – think of it like organizing words in sentences to convey clear ideas. The rule remains: Once the denominators agree, simply add or subtract the numerators as needed. Always ensure that only the tops are juggled once the bases match, and never mess with the shared baseline (denominator).

Simplifying Expressions

With our fractions added, the next step is simplification. Simplification is like cleaning your room – our goal is to tidy up the expression to its simplest, most understandable form.This largely involves combining like terms and clearing mathematical clutter:

• Start by distributing any numbers across combined terms: from \( y(x-4) \), distribute \( y \) to get \( yx - 4y \).
• Now, look for terms you can simplify: from \( 9y + yx - 4y \), simplify to \( yx + 5y \) by combining like terms \(9y - 4y\).

The expression is all set once you remove redundancies. Simplification ensures you have the most efficient form ready for interpretation or further calculations. There are no extra factors here, so you’re done! At its simplest, the solution remains \( \frac{yx + 5y}{x-4} \), clean and ready for further use.