Problem 34 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7}{3 x}-\frac{8}{7 y}-2 $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{49y - 24x - 42xy}{21xy} \).

Step 1: Identify the Operation

The exercise requires you to subtract the rational expressions. The expressions are \( \frac{7}{3x} \) and \( \frac{8}{7y} \), along with a constant \(-2\).

Step 2: Find a Common Denominator

Determine if there's a need for a common denominator to subtract \( \frac{7}{3x} \) and \( \frac{8}{7y} \). Since the denominators are different, a common denominator is \( 21xy \).

Step 3: Adjust the Fractions

Rewrite each fraction with the common denominator of \( 21xy \). For \( \frac{7}{3x} \), multiply the numerator and denominator by \( 7y \): \( \frac{7 \times 7y}{21xy} = \frac{49y}{21xy} \). For \( \frac{8}{7y} \), multiply the numerator and denominator by \( 3x \): \( \frac{8 \times 3x}{21xy} = \frac{24x}{21xy} \).

Step 4: Subtract the Fractions

Combine the two fractions using the common denominator: \( \frac{49y}{21xy} - \frac{24x}{21xy} = \frac{49y - 24x}{21xy} \).

Step 5: Include the Constant

Subtract the constant \(-2\), expressed with the common denominator. Convert \(-2\) to \( \frac{-42xy}{21xy} \) because \(2 \times 21xy = 42xy\).

Step 6: Combine All Terms

Combine all parts: \( \frac{49y - 24x}{21xy} - \frac{42xy}{21xy} = \frac{49y - 24x - 42xy}{21xy} \).

Step 7: Simplify the Expression

The expression \( \frac{49y - 24x - 42xy}{21xy} \) is now in its simplest form as there are no like terms to combine and all terms are accounted for.

Key Concepts

Common Denominator

When dealing with subtraction or addition of fractions, it's crucial to have a shared denominator among the fractions involved. This makes it possible to combine them into a single fraction. In our example, the fractions are \( \frac{7}{3x} \) and \( \frac{8}{7y} \), which clearly have different denominators.

To solve this, we need to find a common denominator, which is essentially a common multiple of the two denominators. Here, the denominators are \( 3x \) and \( 7y \). The simplest way to find a common denominator is to multiply these together.

Thus, our common denominator becomes \( 21xy \). With this common denominator, it's now possible to rewrite both fractions so they can be combined, making computation straightforward.

Subtraction of Fractions

Subtracting fractions may seem tricky at first, but with a few straightforward steps, it becomes manageable. First, write each fraction with the common denominator you identified earlier.

In our case, we rework \( \frac{7}{3x} \) and \( \frac{8}{7y} \) to have \( 21xy \) as their denominator. Let's break it down:

• For \( \frac{7}{3x} \), multiply both the numerator and denominator by \( 7y \) to get \( \frac{49y}{21xy} \).
• For \( \frac{8}{7y} \), multiply both by \( 3x \) to get \( \frac{24x}{21xy} \).

Now that both fractions share the same denominator, we can subtract them directly.

Arrange them as \( \frac{49y}{21xy} - \frac{24x}{21xy} \), then combine the numerators while keeping the common denominator: \( \frac{49y - 24x}{21xy} \).

This method ensures a seamless operation of subtraction over distinct denominators.

Simplification of Algebraic Expressions

Simplifying algebraic expressions is the process of reducing them to their most straightforward form. Once fractions have been combined, check to see if any terms can be combined or reduced further.

In the given problem, after subtraction, we have the expression \( \frac{49y - 24x - 42xy}{21xy} \).

Let's simplify if possible:

• Review the numerator \( 49y - 24x - 42xy \). There are no like terms (terms with the same variable component) here, so there's nothing to combine further.
• If the expression contained any common factor in all terms of the numerator and the denominator, you could divide through by it.

Since no more simplification can be done, our solution remains expressed neatly in its simplest form. Understanding the purpose and method of simplification helps keep our answers clear and concise.

By following gradual simplification, we make complex algebraic expressions more manageable and comprehensible.