Problem 10 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{9}{3 y-21} $$

Step-by-Step Solution

Answer

Question: Simplify the rational expression $$\frac{9}{3y - 21}.$$ Answer: The simplified rational expression is $$\frac{3}{y - 7}.$$

Step 1: Factor out common factors from the numerator and denominator

First, let's find the common factors for both the numerator and denominator: The numerator is \(9\), which can be factored into \(3 \cdot 3\). The denominator is \(3y - 21\), which can be factored by taking out the common factor of \(3\), leading to \(3(y - 7)\). So, the expression becomes: $$ \frac{9}{3y - 21} = \frac{3 \cdot 3}{3(y - 7)} $$

Step 2: Cancel out the common factors

Now, we can cancel out the common factor of \(3\) from both the numerator and the denominator: $$ \frac{3 \cdot 3}{3(y - 7)} = \frac{3}{(y - 7)} $$ Now the expression has been reduced to its lowest terms: $$ \frac{9}{3y - 21} = \frac{3}{y - 7} $$

Key Concepts

Factoring Algebraic Expressions

Factoring algebraic expressions is akin to breaking down numbers into their basic building blocks; it's a critical step when simplifying complex algebraic expressions. When we factor, we look for common elements in terms that can be pulled out, making the equation simpler and more manageable.

For instance, consider the denominator of the textbook example, which is: \(3y - 21\). To factor this expression, we identify the greatest common factor, which here is \(3\). We then divide each term by \(3\) to get \(y - 7\), simplifying the expression to \(3(y - 7)\). Factoring is a powerful tool and serves as a prelude to further simplification steps like canceling.

Canceling Common Factors

Once we have an expression factored, we can move to canceling common factors. This is the process where the same number or variable that appears in both the numerator and the denominator can be removed, thus simplifying the expression further.

In our exercise, we have \(3 \cdot 3\) in the numerator and \(3(y - 7)\) in the denominator. Since \(3\) is common to both, we can cancel it out. It's as if they nullify each other, leaving us with a cleaner, simpler expression: \(\frac{3}{y - 7}\). Remember, we can only cancel factors that are multiplied, not those that are added or subtracted. Canceling common factors reduces the complexity of expressions and can often reveal a more straightforward relationship between variables.

Reducing Fractions to Lowest Terms

Reducing fractions to their lowest terms makes them easier to work with and understand. This process involves factoring the numerator and the denominator and then canceling out all common factors until no more simplification is possible.

In our textbook example, after canceling the \(3\), we end up with \(\frac{3}{y - 7}\). This final expression is now in its lowest terms because there are no common factors left to cancel out. In general, a fraction is in its lowest terms when the only common factor between the numerator and the denominator is \(1\). Simplifying expressions to their lowest terms not only makes calculations easier but also helps in clearly understanding the underlying mathematical relationships.