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Chapter 7: Problem 82

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7 x}{x^{2}-y^{2}}-\frac{3}{y-x}$$

Short Answer

Expert verified
Solving this exercise results in \(-\frac{7x - 3(x + y)}{(x - y)(x + y)}\), which can be further simplified for your final solution.

Step by step solution

01

Rewrite the Second Fraction

Notice that 'y - x' can be rewritten as '-(x - y)'. So rewrite the second fraction as \(-\frac{3}{x - y}\). Now the exercise should look like this: \(\frac{7x}{x^{2}-y^{2}}- (-\frac{3}{x - y})\)
02

Simplify and Factorize the First Fraction's Denominator

Next, simplify the denominator of the first fraction using difference of squares, a formula in algebra (i.e, \(a^2 - b^2 = (a - b)(a + b)\)). So \(x^{2} - y^{2}\) becomes \((x - y)(x + y)\). Now the first fraction looks like this: \(\frac{7x}{(x - y)(x + y)}\)
03

Make Denominators Identical and Perform the Subtraction

Now the denominators are \((x - y)(x + y)\) and \(x - y\) respectively. Make the denominators identical by multiplying the second fraction by \((x + y)\) both on top and bottom, to get \[-\frac{3}{x - y} \cdot \frac{x + y}{x + y}\]. Now perform the subtraction.
04

Simplify the Result

Simplify the result in the numerator by distributing and combining like terms, if any.

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