Problem 45 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 y}{4 y-20}+\frac{9 y}{6 y-30}$$

Step-by-Step Solution

Answer

The simplified result of the addition is \(\frac{27y}{12(y-5)}\).

Step 1: Find the Least Common Denominator (LCD)

The denominators of the rational expressions are \(4 y - 20\) and \(6 y - 30\) respectively. Simplify each denominator by factoring out the common factor: \(4 y - 20 = 4 (y - 5)\) and \(6 y - 30 = 6 (y - 5)\). As a result, the LCD is \(12 (y - 5)\).

Step 2: Rewrite each fraction with the LCD

Now each fraction is expressible in terms of the LCD and can be written as follows: \(\frac{3y}{4(y - 5)} = \frac{3y \cdot 3}{3 \cdot 4 (y - 5)} = \frac{9y}{12(y-5)}\) and \(\frac{9y}{6(y - 5)} = \frac{9y \cdot 2}{2 \cdot 6 (y - 5)} = \frac{18y}{12(y-5)}\).

Step 3: Perform the Addition

We are now in a position to add the two fractions. Thus, \(\frac{9y}{12(y-5)} + \frac{18y}{12(y-5)} = \frac{9y + 18y}{12(y-5)}\).

Step 4: Simplify the Result

Simplifying the numerator of the resulting fraction gives \(\frac{27y}{12(y-5)}\). This expression cannot be simplified further.