Problem 70 - Introductory Algebra for College Students

Question

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{y}{a x+b x-a y-b y}-\frac{x}{a x+b x-a y-b y}$$

Step-by-Step Solution

Answer

The result of the indicated operation is \(\frac{y-x}{a x+b x-a y-b y}\).

Step 1: Identify the structure

The fractions \(\frac{y}{a x+b x-a y-b y}\) and \(\frac{x}{a x+b x-a y-b y}\) share the same denominator \(a x+b x-a y-b y\). This means that they can be subtracted directly.

Step 2: Subtract the numerators

With the knowledge that the denominators are the same, the operation can proceed by subtracting the numerators. Hence, subtraction of the two complex fractions simplifies to \(\frac{y-x}{a x+b x-a y-b y }\).

Step 3: Present the final result

The subtraction of the two fractions reduces to \(\frac{y-x}{a x+b x-a y-b y}\). There's no possible simplification in this context, because the expression in the numerator doesn't have common terms with the denominator.

Key Concepts

Subtracting Fractions in Algebra

When working with algebraic expressions that include fractions, it's common to encounter problems that require subtracting one fraction from another.
Understanding this process is critical for simplifying complex algebraic expressions. Here's a simplified explanation:

• Ensure each fraction has the same denominator. If they don't, find a common denominator.
• Once the denominators are the same, subtract the numerators of the fractions. The denominator remains unchanged.
• Simplify the resulting fraction if possible by factoring and canceling out common factors.

As seen in the exercise, since both fractions already had a common denominator, the problem could directly proceed to subtract the numerators, leading to the simplified form \(\frac{y-x}{a x+b x-a y-b y}\).

Finding Common Denominators in Algebra

A key step in adding or subtracting algebraic fractions is ensuring they share a common denominator.
Here’s how to find one:

• For simple expressions, the least common denominator (LCD) might just be the product of the individual denominators.
• With more complex or similar denominators, you may simplify them first or factor out common terms to find the LCD.
• Once the LCD is found, each fraction is adjusted so that it has this denominator. This may involve multiplying the numerator and denominator of each fraction by necessary factors.

This process facilitates the combination of fractions into a single, simplified expression. The example given in the original exercise didn't require this step since the denominators were already identical— a convenience that won't always be present.

Simplifying Complex Fractions

Complex fractions often intimidate students, but they can be tackled by breaking down the simplification process into manageable steps.
Follow these guidelines:

• Firstly, identify any common factors in the numerator and denominator.
• Then, simplify the numerator and the denominator separately as much as possible. This might include factoring polynomials or canceling out like terms.
• If applicable, perform any addition or subtraction in the numerator or denominator to combine into a single fraction.
• Lastly, if the complex fraction can be simplified further, divide the numerator by the denominator or multiply by the reciprocal of the denominator.

The exercise presented didn't yield further simplification because the numerator and denominator didn't share common factors. Hence, the solution remained as \(\frac{y-x}{a x+b x-a y-b y}\). When faced with your own complex fractions, remember to seek out these simplifications to make the expressions easier to work with.