Problem 60 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{5 y^{2}-5 y}-\frac{2}{5 y-5}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{7-2y}{5y(y-1)}\).

Step 1: Identify the Least Common Denominator (LCD).

Observe the denominators \(5 y^{2}-5 y\) and \(5 y-5\). Notice that we can factor out 5 and also 'y' from the first term of both denominators:So, the denominators becomes \(5y(y-1)\) and \(5(y-1)\) respectively. Now, it's evident that the LCD is \(5y(y-1)\) since it covers all the individual terms in the two denominators.

Step 2: Rewrite the fractions using the LCD.

With the identified LCD as \(5y(y-1)\), we write each fraction over this denominator:The first fraction does not change, while the second fraction needs to be multiplied by 'y' on both the numerator and denominator to match the LCD. This gives us \(\frac{7}{5y(y-1)} - \frac{2y}{5y(y-1)}\).

Step 3: Subtract the numerators.

Being under the same denominator now, we can perform the subtraction on the numerators only, this gives us: \(\frac{7-2y}{5y(y-1)}\).

Step 4: Simplify the result.

In this step, check if the obtained fraction can be simplified further. However, neither the numerator nor the denominator can be factored so the current result is already in its simplest form.