Problem 17 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{y+9}{y^{2}+9 y} $$

Step-by-Step Solution

Answer

\frac{1}{y}

Step 1 - Identify the Common Factor

Look at the numerator and the denominator of the fraction. Identify any common factors. In this case, both terms in the denominator have a common factor of \( y \).

Step 2 - Factor the Common Factor Out

Factor out the common factor in the denominator: \(\frac{y+9}{y(y+9)}\)

Step 3 - Simplify the Fraction

Now, cancel out the common factor \( y+9 \) in both the numerator and the denominator: \(\frac{y+9}{y(y+9)} = \frac{1}{y}\).

Key Concepts

Common Factor

A common factor is a number or term that divides both the numerator and the denominator without leaving a remainder. In our exercise, finding the common factor is the first step. Here, the denominator is composed of two terms: \(y^2\) and \(9y\). Both these terms have 'y' as a factor. Identifying and factoring out the common factor helps simplify the equation. Always start by looking at each term in the numerator and the denominator to discover these common factors.

Factoring

Factoring involves breaking down a number or expression into its component parts or 'factors'. In the exercise above, the denominator \(y^2 + 9y\) can be factored by taking out the common factor \(y\). This means rewriting \(y^2 + 9y\) as \(y(y + 9)\). This process makes it easier to handle and simplify the expression. By factoring out common terms, complex fractions become manageable. It’s like peeling layers off to make simplifying easier.

Canceling Terms

After factoring, the next step is to cancel out common terms from both the numerator and the denominator. When both parts of a fraction share a common factor, you can remove it. In our example, the fraction becomes \(\frac{y+9}{y(y+9)}\). Here, \(y+9\) is present in both the numerator and the denominator. By canceling \(y+9\), you simplify the fraction to \(\frac{1}{y}\). Canceling common terms is crucial for simplifying the expression to its most reduced form.