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Chapter 7: Problem 81

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 x}{x^{2}-y^{2}}-\frac{2}{y-x}$$

Short Answer

Expert verified
The simplified expression is \(\frac{3x + 2}{(x+y)(x-y)}\).

Step by step solution

01

Factor The Difference of Squares

Factoring the denominator in the first expression, we get \(x^{2}-y^{2} = (x+y)(x-y)\). This gives the new expression: \(\frac{3x}{(x+y)(x-y)}-\frac{2}{y-x}\).
02

Make Denominators Same

Before we can combine the fractions, they need to have the same denominator. Notice that \(x-y\) and \(y-x\) are opposite in sign. Therefore we can rewrite \(y-x\) as \(-(x-y)\) to match the denominator in the first expression. Now the expressions read as \(\frac{3x}{(x+y)(x-y)}+\frac{2}{-(x-y)}.\)
03

Combine the Fractions

Since both fractions now have the same denominator, we combine the numerators over their common denominator: \(\frac{3x - 2}{(x+y)(x-y)}\). However, this expression is equal to \(\frac{3x + 2}{(x+y)(x-y)}\) because when the numerator and denominator of a fraction are divided by -1, the fraction remains the same due to reducing -1/-1 to 1.
04

Final Simplified Expression

There are no common factors in the numerator and the denominator that would simplify further. Therefore the final simplified expression is \(\frac{3x + 2}{(x+y)(x-y)}\).

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Most popular questions from this chapter

add or subtract as indicated. Simplify the result, if possible. $$\begin{aligned} &\frac{x^{2}+9 x}{4 x^{2}-11 x-3}+\frac{3 x-5 x^{2}}{4 x^{2}-11 x-3}\\\ &x^{2}-4 x-4 x-4 \end{aligned}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-2}+\frac{4}{2-x}$$

What is the relationship among time traveled, distance traveled, and rate of travel?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used \(\frac{a}{d}=\frac{b}{e}\) to show that corresponding sides of similar triangles are proportional, but I could also use \(\frac{a}{b}=\frac{d}{e}\) or \(\frac{d}{a}=\frac{e}{b}\)

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}-13}{x+4}-\frac{3}{x+4}=x+4, x \neq-4$$
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