Problem 82 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \(\frac{7x}{x^{2}-y^{2}}- (-\frac{3}{x - y})\)

Step 2: 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: \(\frac{7x}{(x - y)(x + y)}\)

Step 3: 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.

Step 4: Simplify the Result

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