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Chapter 7: Problem 63

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

Short Answer

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

Step by step solution

01

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.
02

Find the Common Denominator

The common denominator between the two fractions is (y-2)(y+5).
03

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 \(\frac{(y-7)(y+5)}{(y-2)(y+5)}\)
04

Perform the Subtraction

Now perform the subtraction \(\frac{y^{2}-39}{(y-2)(y+5)} - \frac{(y-7)(y+5)}{(y-2)(y+5)} = \frac{y^{2}-39-(y^{2}-2y-35)}{(y-2)(y+5)}\)
05

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 \(\frac{2y-4}{(y-2)(y+5)}\)

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Most popular questions from this chapter

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. $$\frac{x^{2}+4 x+3}{x+2}-\frac{5 x+9}{x+2}=x-2, x \neq-2$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x-5}{6} \cdot \frac{3}{5-x}=\frac{1}{2}\( for any value of \)x$ except 5$$

Solve: \(x^{2}-12 x+36=0 .\) (Section 6.6, Example 4)

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