Problem 33 - College Algebra

Question

Add and subtract the rational expressions, and then simplify. $$ \frac{4}{x}+\frac{10}{y} $$

Step-by-Step Solution

Answer

\(\frac{4y + 10x}{xy}\) is the simplified expression.

Step 1: Identify the Denominators

The denominators of the two rational expressions are \(x\) and \(y\). To combine them, we need a common denominator. Our task is to find the least common denominator (LCD) for these fractions.

Step 2: Find the Least Common Denominator

Since \(x\) and \(y\) are different variables, the least common denominator will be their product, \(xy\). This allows us to rewrite each fraction with the common denominator \(xy\).

Step 3: Rewrite Each Fraction with the LCD

Convert the first fraction: \(\frac{4}{x} = \frac{4y}{xy}\). Convert the second fraction: \(\frac{10}{y} = \frac{10x}{xy}\). Now both fractions have \(xy\) as their denominator.

Step 4: Add the Fractions

Add the numerators together over the common denominator: \(\frac{4y}{xy} + \frac{10x}{xy} = \frac{4y + 10x}{xy}\).

Step 5: Simplify the Expression

The expression \(\frac{4y + 10x}{xy}\) can't be simplified further because 4 and 10 have no common factors with \(x\) or \(y\). Thus, the simplified form is \(\frac{4y + 10x}{xy}\).

Key Concepts

Least Common Denominator

When adding or subtracting fractions, finding a common denominator is crucial. For rational expressions, the common denominator is known as the Least Common Denominator (LCD). It allows you to express fractions in a way that makes them easier to add or subtract.

Imagine if you were combining two different containers of liquid, you would first need a bigger container to pour both into. Similarly, the LCD acts as the bigger container, accommodating both fractions simultaneously.

• Identify the denominators of the expressions involved.
• Determine if they have any common factors.
• If not, multiply the denominators to find the LCD.

In the given problem, we are working with denominators \(x\) and \(y\). Since these are different variables with no common factors, their product \(xy\) serves as the LCD. By making the denominators the same, we can then proceed to add the numerators easily.

Adding Fractions

Once the least common denominator is identified, the next step is adding fractions, which involves aligning them so both have this common denominator. This alignment is what allows you to simply add their numerators.

Here's the process broken down:

• Rewrite each fraction using the LCD as the new denominator.
• Adjust the numerators to reflect this change.
• Add the adjusted numerators.

In our example,

• First fraction \(\frac{4}{x}\) becomes \(\frac{4y}{xy}\).
• Second fraction \(\frac{10}{y}\) becomes \(\frac{10x}{xy}\).

With both fractions transformed to have the same denominator \(xy\), we can sum them up by adding the numerators: \(4y + 10x\). This results in the fraction \(\frac{4y + 10x}{xy}\). This method ensures that no aspect of the fractions is lost in the addition process.

Simplifying Expressions

After adding the fractions, the next step is usually to simplify expressions if possible. Simplifying helps in making the expressions more manageable and easier to understand. However, not every expression can be simplified after addition.

Here's what to do:

• Check if the terms in the numerator and denominator can be factored out further.
• Determine if there are common factors that can be cancelled.
• If simplification isn't possible, confirm the expression is in its simplest form.

In this exercise, the resulting expression after addition is \(\frac{4y + 10x}{xy}\). Upon reviewing the numerator (\(4y + 10x\)), there are no common factors that can simplify the fraction further with respect to the denominator \(xy\). Thus, this expression is already in its simplest form. It's vital to understand that sometimes simplification is not feasible, and recognizing when this is the case is an important part of working with rational expressions.