Problem 74 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-7}{3 y^{2}}-\frac{y-2}{12 y}$$

Step-by-Step Solution

Answer

The simplified result of the fraction subtraction is \(\frac{-y^{2} + 6y - 28}{12 y^{2}}\).

Step 1: Find a common denominator

To subtract fractions, they must have the same denominator. In these two fractions, the denominators are \(3 y^{2}\) and \(12 y\). The common denominator here is \(12 y^{2}\). So, the first term remains as is, while the second term must be multiplied by \(y\) top and bottom to align it with the common denominator.

Step 2: Rewrite the fractions with the common denominator

Using the common denominator, \(12 y^{2}\), the fractions can be rewritten as follows: \(\frac{4(y-7)}{12 y^{2}}-\frac{y(y-2)}{12 y^{2}}\). This has been done by multiplying the second term by \(y\) both in the numerator and denominator.

Step 3: Perform the subtraction

Now that the fractions have the same denominator, subtract the numerators: \(\frac{4(y-7)- y(y-2)}{12 y^{2}}\).

Step 4: Expand the numerator

Expand the numerators and simplify: \(\frac{4y - 28 - y^{2} + 2y}{12 y^{2}} = \frac{-y^{2} + 6y - 28}{12 y^{2}}\).

Step 5: Final Simplification

This fraction cannot be simplified further because the numerator and the denominator do not have common factors. So the final answer is \(\frac{-y^{2} + 6y - 28}{12 y^{2}}\).