Problem 59 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{5}{2 y^{2}-2 y}-\frac{3}{2 y-2}$$

Step-by-Step Solution

Answer

The simplified result of the expression \(\frac{5}{2 y^{2}-2 y}-\frac{3}{2 y-2}\) is \(\frac{-3y+5}{2y(y-1)}\).

Step 1: Determine the Least Common Denominator (LCD)

Observe the denominators of the fractions: \(2 y^{2}-2 y\) and \(2 y-2\). Derive a common factor from each polynomial and express it in factored form. Thus, we have: \(2 y^{2}-2 y = 2y(y-1)\) and \(2 y-2 = 2(y-1)\). Hence, the LCD is \(2y(y-1)\).

Step 2: Rewrite the fractions with the LCD

Rewrite each fraction as an equivalent fraction with the LCD as the denominator: \(\frac{5}{2 y^{2}-2 y}=\frac{5}{2y(y-1)}\) and \(\frac{3}{2 y-2}=\frac{3y}{2y(y-1)}\).

Step 3: Subtract the fractions

Subtract the two fractions that now have the same denominator, \(\frac{5}{2y(y-1)} - \frac{3y}{2y(y-1)} = \frac{5-3y}{2y(y-1)}\). Note that the subtraction happens only with the numerators while the denominator remains the same.

Step 4: Simplify the result

Simplify the fraction \(\frac{5-3y}{2y(y-1)}\) by ordering the terms in the numerator from highest to lowest power of y: \(\frac{-3y+5}{2y(y-1)}\). The final result cannot be further simplified and therefore it is the final answer.