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Chapter 6: Problem 41

Perform the indicated operations and simplify. For Exercises \(33,34,39,\) and \(40,\) check the solution with a graphing calculator. $$\frac{\frac{x}{y}-\frac{y}{x}}{1+\frac{y}{x}}$$

Short Answer

Expert verified
The simplified form is \( \frac{x-y}{y} \).

Step by step solution

01

Rewrite Complex Fractions

Begin by rewriting the given expression to make it easier to simplify. The complex fraction is: \( \frac{\frac{x}{y}-\frac{y}{x}}{1+\frac{y}{x}} \). This can be rewritten as two separate parts: the numerator \( \frac{x}{y} - \frac{y}{x} \) and the denominator \( 1 + \frac{y}{x} \).
02

Find a Common Denominator for the Numerator

In the numerator \( \frac{x}{y} - \frac{y}{x} \), find a common denominator for the two fractions. The common denominator is \( yx \). Rewrite each term: \( \frac{x}{y} = \frac{x^2}{yx} \) and \( \frac{y}{x} = \frac{y^2}{yx} \). Thus, the numerator becomes: \( \frac{x^2 - y^2}{yx} \).
03

Simplify the Denominator

The denominator is \( 1 + \frac{y}{x} \). Rewrite this to have a common denominator: \( \frac{x}{x} + \frac{y}{x} = \frac{x + y}{x} \).
04

Divide the Fractions

Now that we have the numerator \( \frac{x^2 - y^2}{yx} \) and the denominator \( \frac{x + y}{x} \), rewrite the given complex fraction: \( \frac{\frac{x^2 - y^2}{yx}}{\frac{x + y}{x}} \). To divide the fractions, multiply by the reciprocal of the denominator: \( \frac{x^2 - y^2}{yx} \times \frac{x}{x + y} \).
05

Simplify the Expression

Cancel common terms in the fractions: \( x \) in the numerator and \( x \) in the denominator, and \( x + y \) in the denominator. The simplified form becomes: \( \frac{x^2 - y^2}{y(x + y)} \).
06

Recognize the Difference of Squares

Note that the term \( x^2 - y^2 \) in the numerator is a difference of squares, which can be factored: \( (x-y)(x+y) \). Substitute back into the expression: \( \frac{(x-y)(x+y)}{y(x + y)} \).
07

Simplify Further if Possible

Cancel \( (x+y) \) from the numerator and the denominator. The final simplified expression is \( \frac{x-y}{y} \).
08

Verification with a Graphing Calculator

To verify the solution, input the original expression \( \frac{\frac{x}{y} - \frac{y}{x}}{1 + \frac{y}{x}} \) and the simplified expression \( \frac{x-y}{y} \) into a graphing calculator. Both should yield the same graphical representation and table values for various numerical values of \( x \) and \( y \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Fractions
Complex fractions can seem a bit intimidating at first glance, but they are simply fractions within fractions. Imagine a fraction as a sandwich, and complex fractions are like a sandwich with multiple layers. The exercise we鈥檙e looking at has the complex fraction:
  • The numerator: \(\frac{x}{y} - \frac{y}{x}\)
  • The denominator: \(1 + \frac{y}{x}\)
By breaking the complex fraction down into its constituent parts, solving becomes more manageable. The goal is to rewrite everything in terms of a common denominator. This makes the fractions easier to handle and prepares them for further simplification. Think of it as aligning all ingredients on a sandwich to make it balanced and tasty.
Difference of Squares
A crucial algebraic identity is the difference of squares, written as \(a^2 - b^2 = (a-b)(a+b)\). It's a gem of algebra that allows us to factor certain expressions beautifully. In our exercise, you鈥檒l notice the numerator is \(x^2 - y^2\). This fits perfectly with the difference of squares formula, making it quickly recognizable and factorizable.
  • Recognize the pattern: \(x^2 - y^2\)
  • Apply the formula: \((x-y)(x+y)\)
This transformation simplifies processes and helps in canceling out terms in subsequent steps. Discover the magic of seeing such patterns鈥攐nce you spot it, simplifying becomes a walk in the park! The difference of squares is a frequent guest in algebra problems, so getting comfortable with it is a big confidence booster.
Graphing Calculators
Graphing calculators are fantastic tools in a mathematician鈥檚 toolkit. They not only help in checking solutions visually but are also great for understanding the behavior of functions. In our problem, the graphing calculator acts like a buddy who helps verify our calculations.
  • Graph the original and simplified expression
  • Check if they overlap perfectly
If both graphs match, it鈥檚 a green signal. Additionally, they provide table values for various \(x\) and \(y\) to add further confirmation. These devices ensure that not only do you understand each step, but you also gain a visual insight into how functions behave in different scenarios. Plus, it makes learning interactive and fun!
Simplification Steps
The process of simplification is like decluttering a messy room. Initially, the pile of algebraic expressions might seem overwhelming, but taking it step-by-step transforms chaos into order. Here鈥檚 a recap of how simplification works like a charm:
  • Step 1: Separate and rewrite into simpler fractions
  • Step 2: Find a common denominator
  • Step 3: Simplify or factor the terms wherever applicable utilizing identities
  • Step 4: Cancel out common terms across numerator and denominator
Each step-in simplification, like removing common terms, tidies up the expression. The aim is to simplify so much that you can see the expression鈥檚 simpler, clearer form. This exercise鈥檚 target was to reach \(\frac{x-y}{y}\). By following simplification steps diligently, you navigate from complexity to simplicity with elegance!

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Most popular questions from this chapter

Use the special products of this section to determine the products. You may need to write down one or two intermediate steps. $$(x+y)^{2}(x-y)^{2}$$

Factor the given expressions completely. $$12 m^{2}+60 m+75$$

Solve the given problems. The length of a piece of rectangular floor tile is \(3 \mathrm{cm}\) more than twice the side \(x\) of a second square piece of tile. The width of the rectangular piece is \(3 \mathrm{cm}\) less than twice the side of the square piece. Find the area of the rectangular piece in terms of \(x\) (in expanded form).

Solve the given problems. If \((x+k)(x-1)=x^{2}+k(x+2)-(x-3),\) find the value of \(k\).

Find constants A and B such that the equation is true. $$\frac{x-12}{x^{2}+x-6}=\frac{A}{x+3}+\frac{B}{x-2}$$
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