Problem 81 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \(\frac{3x}{(x+y)(x-y)}-\frac{2}{y-x}\).

Step 2: 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 \(\frac{3x}{(x+y)(x-y)}+\frac{2}{-(x-y)}.\)

Step 3: 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.

Step 4: Final Simplified Expression

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