Problem 46 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{4 y}{5 y-10}+\frac{3 y}{10 y-20}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{11y}{10(y-2)} \)

Step 1: Simplify the Denominators

The denominators have a common factor that can be factored out. For 5y-10, we factor out 5, to get 5(y-2). For 10y-20, we factor out 10 to get 10(y-2). So, \( \frac{4 y}{5 y-10}+\frac{3 y}{10 y-20} \) can be written as \( \frac{4 y}{5(y-2)}+\frac{3 y}{10(y-2)} \)

Step 2: Make the Denominators the Same

The denominators in both fractions should be the same in order to add the fractions. We can see that, the denominator of the second fraction is twice the first one. So, we multiply the numerator and the denominator of the first fraction by 2 to make the denominators equal. As a result, our expression is now \( \frac{8 y}{10(y-2)}+\frac{3 y}{10(y-2)} \)

Step 3: Add the Fractions

Now that the denominators are the same, we can add the two fractions. The result is \( \frac{8y+3y}{10(y-2)} \)

Step 4: Combine Like Terms

Add 8y and 3y in the numerator to get 11y. So the final expression is \( \frac{11y}{10(y-2)} \)