Problem 63 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y^{2}-39}{y^{2}+3 y-10}-\frac{y-7}{y-2}$$

Step-by-Step Solution

Answer

The simplified result of subtraction is \(\frac{2y-4}{(y-2)(y+5)}\)

Step 1: Factorize the Denominators

To find the common denominator, both denominators have to be factored first. The denominator \(y^{2}+3 y-10\) in the first fraction can be factored into (y-2)(y +5) and the denominator of the second fraction is already factored as y-2.

Step 2: Find the Common Denominator

The common denominator between the two fractions is (y-2)(y+5).

Step 3: Rewrite the Fractions with the Common Denominator

Rewrite the fractions using the common denominator. Here, the first fraction remains unchanged as \(\frac{y^{2}-39}{(y-2)(y+5)}\). The denominator of the second fraction is multiplied by (y+5) to obtain the common denominator, thus becomes \(\frac{(y-7)(y+5)}{(y-2)(y+5)}\)

Step 4: Perform the Subtraction

Now perform the subtraction \(\frac{y^{2}-39}{(y-2)(y+5)} - \frac{(y-7)(y+5)}{(y-2)(y+5)} = \frac{y^{2}-39-(y^{2}-2y-35)}{(y-2)(y+5)}\)

Step 5: Simplify the Result

Simplify the fraction by expanding and cancelling: \(\frac{y^{2}-39-(y^{2}-2y-35)}{(y-2)(y+5)}\) becomes\(\frac{y^{2}-39-y^{2}+2y+35}{(y-2)(y+5)}\) which simplifies to \(\frac{2y-4}{(y-2)(y+5)}\)