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Chapter 7: Problem 73

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y+3}{5 y^{2}}-\frac{y-5}{15 y}$$

Short Answer

Expert verified
The simplified result of the given expression is \(\frac{2(y + 7)}{15y}\).

Step by step solution

01

Identify the common denominator

In this expression, the two denominators are \(5y^{2}\) and \(15y\). A common denominator for these two fractions is their least common multiple (LCM). A multiple of \(5y^{2}\) can be written as \(5y^2 脳 m\), and a multiple of \(15y\) can be written as \(15y 脳 n\). The LCM is the smallest value for which \(m\) and \(n\) are whole numbers. In this case, the LCM (and thus the common denominator) is \(15y^{2}\).
02

Rewrite the fractions

We need to rewrite each fraction so they both have the denominator \(15y^{2}\). The first fraction becomes \(\frac{(y+3) \times 3y}{15y^{2}}\) and the second fraction becomes \(\frac{(y-5) 脳 y}{15y^{2}}\). This results in the new expression: \(\frac{3y(y+3)}{15y^{2}} - \frac{y(y-5)}{15y^{2}}\).
03

Combine like terms

Now that the fractions have the same denominator, they can be subtracted. This results in: \(\frac{3y(y+3) - y(y-5)}{15y^{2}}\).
04

Distribute and simplify

Next, you distribute the \(3y\) and \(-y\) to terms inside the brackets to get: \(\frac{3y^2+9y - y^2 +5y}{15y^{2}}\). This simplifies to \(\frac{2y^{2} + 14y}{15y^{2}}\). This simplifies further by dividing through by \(y\) (given that \(y \neq 0\)) to get: \(\frac{2y + 14}{15y}\).
05

Simplify further

The numerator has common factors that can be factored out which gives \(\frac{2(y + 7)}{15y}\). This is the simplest form of the original expression given.

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