Problem 89 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{9 y+3}{y^{2}-y-6}+\frac{y}{3-y}+\frac{y-1}{y+2}$$

Step-by-Step Solution

Answer

The result of the operation is \( \frac{-2y^2+10y+6}{(y-3)(y+2)} \)

Step 1: Factorize the Denominators

Factorize the denominators of the fractions to find a common denominator:\[\frac{9 y+3}{(y-3)(y+2)}+\frac{y}{3-y}+\frac{y-1}{y+2}\]Note: \(3-y\) is equal to \(-(y-3)\). Therefore, the second fraction can be written as:\[\frac{9 y+3}{(y-3)(y+2)}-\frac{y}{y-3}+\frac{y-1}{y+2}\]

Step 2: Find a Common Denominator and Rewrite the Fractions

Find a common denominator, which is \((y-3)(y+2)\). Rewrite the fractions so they all have this common denominator:\[\frac{9 y+3}{(y-3)(y+2)}-\frac{(y)(y+2)}{(y-3)(y+2)}+\frac{(y-1)(y-3)}{(y-3)(y+2)}\]

Step 3: Combine Like Terms

Add and subtract the numerators over the common denominator:\[\frac{9y+3-y^2-2y-y^2+3y+3}{(y-3)(y+2)}\]Which simplifies to:\[\frac{-2y^2+10y+6}{(y-3)(y+2)}\]

Step 4: Simplify the Result

The result is a simple fraction which is the solution:\[\frac{-2y^2+10y+6}{(y-3)(y+2)}\]