Problem 40 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{x}{y}+1}{1-\frac{x}{y}} $$

Step-by-Step Solution

Answer

The simplified form of the fraction is \(\frac{x+y}{y-x}\).

Step 1: Rewrite the Complex Fraction

The given complex fraction is \(\frac{\frac{x}{y}+1}{1-\frac{x}{y}}\). Start by rewriting this expression by having a single fraction in the numerator and a single fraction in the denominator. The numerator can be rewritten such that \(\frac{x}{y} + 1\) becomes \(\frac{x+y}{y}\), and the denominator \(1-\frac{x}{y}\) becomes \(\frac{y-x}{y}\). This gives us the new expression as \(\frac{\frac{x+y}{y}}{\frac{y-x}{y}}\).

Step 2: Simplify by Division

Now that we have a single fraction in both the numerator and the denominator, simplify the complex fraction by dividing the fraction in the numerator by the fraction in the denominator. This leads to \(\frac{x+y}{y}\times \frac{y}{y-x}\).

Step 3: Simplify the Expression

Multiply the fractions from Step 2 to get the simplified result. Therefore, \(\frac{x+y}{y}\times \frac{y}{y-x} = \frac{(x+y) \cdot y}{y \cdot (y-x)}\). The \(y\) in the numerator and the denominator cancels out, leaving us with \(\frac{x+y}{y-x}\).

Key Concepts

Fraction Simplification

Complex fractions, like the one presented in the exercise, can seem daunting at first. However, with the right approach, they become quite manageable. Fraction simplification is essential in mathematics because it allows us to reduce fractions to their simplest form, making complex problems easier to handle.

In this specific exercise, the complex fraction \( \frac{\frac{x}{y} + 1}{1 - \frac{x}{y}} \) was simplified by first rewriting each part as a single fraction. This process involved:

• Turning the fraction in the numerator \( \frac{x}{y} + 1 \) into \( \frac{x+y}{y} \)
• Converting the denominator \( 1 - \frac{x}{y} \) to \( \frac{y-x}{y} \)

This made the expression more uniform, setting the stage for easy simplification. Sometimes, simplification involves canceling out common factors, as seen when \( y \) was canceled in the final steps, resulting in \( \frac{x+y}{y-x} \). Ensuring each step is clear helps to avoid confusion and errors.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. In intermediate algebra, it's important to understand how to manipulate these expressions for simplification and equation solving.

The exercise you tackled today involved rational expressions such as \( \frac{x}{y} \). These expressions were combined and manipulated step by step to bring clarity.

The core method used was multiplying by the reciprocal of the fraction in the denominator.

• Consider the expression \( \frac{\frac{x+y}{y}}{\frac{y-x}{y}} \).
• To simplify this, you multiply by the reciprocal, converting it to \( \frac{x+y}{y} \times \frac{y}{y-x} \).

This step leverages the multiplication of fractions, allowing simplification through factor cancellation. Understanding these principles aids in grasping more intricate rational expression manipulations.

Intermediate Algebra

Intermediate algebra builds upon foundational concepts such as simplifying fractions and working with rational expressions. In this context, dealing with complex fractions as we did in the exercise requires an understanding of manipulation techniques including rewriting and simplifying expressions.

Key to mastering intermediate algebra is the ability to recognize patterns and apply the correct operations effectively. Intermediate algebra often involves processes such as:

• Simplifying fractions by combining like terms.
• Using reciprocal and factoring techniques.
• Managing polynomial expressions and equations through strategic optimization.

By practicing these methods regularly, you not only improve your algebraic skills but also prepare for more advanced topics. Always ensure each step is logically sound and keeps the core principles of algebra in focus. In this exercise, the step-by-step simplification helped to reinforce these concepts, leading us to a clear and simplified expression.