Problem 9 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{12 x y}{42 y}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2x}{7}\).

Step 1: Identify Common Factors

First, look for common factors in the numerator and the denominator of the given rational expression. The numerator is \(12xy\), and the denominator is \(42y\).

Step 2: Factorize Numbers

Factorize the numerical coefficients in both the numerator and the denominator. 12 can be factored into \(2 \times 2 \times 3\) and 42 can be factored into \(2 \times 3 \times 7\).

Step 3: Cancel Common Factors

Identify and cancel the common factors in the numerator and the denominator. We have the common factors \(2\) and \(3\). After canceling these common factors, the rational expression simplifies as follows: \(\frac{12xy}{42y} = \frac{4xy}{14y}\).

Step 4: Simplify by Canceling Variables

Since there is a common factor of \(y\) in both the numerator and the denominator, we can cancel \(y\) to further simplify the expression: \(\frac{4xy}{14y} = \frac{4x}{14}\).

Step 5: Simplify the Remaining Coefficients

Now, simplify \(\frac{4x}{14}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. This gives \(\frac{2x}{7}\) as the final simplified expression.

Key Concepts

Factoring

When you need to simplify a rational expression, factoring is often your first step. The goal here is to break down the numbers and variables in both the numerator and denominator into their simplest components. This means expressing them as a product of their factors. For example: - Consider the expression in the numerator \(12xy\), which consists of the number 12 and the variable products \(x\) and \(y\).- Focus first on the number 12. This can be factored into prime numbers as \(2 \times 2 \times 3\).
Now, let's apply the same logic to the denominator, 42. By factoring 42, we break it down into prime factors as \(2 \times 3 \times 7\).
Understanding how to factor expressions allows you to simplify expressions by identifying any common factors that the numerator and the denominator share.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is critical in simplifying expressions.It is the largest number that divides two or more numbers without leaving a remainder. In the context of simplifying rational expressions, knowing the GCD helps determine which factors can be canceled out. For our example, let's look at the factored numerical coefficients from both the numerator and denominator: - The number 12 is factored to \(2 \times 2 \times 3\).- The number 42 is factored to \(2 \times 3 \times 7\).
The GCD of 12 and 42 is important because it tells us exactly which numbers these two values have in common. Here, both numbers have the factors 2 and 3 in common. Therefore, the GCD is \(2 \times 3 = 6\).Knowing the GCD is helpful in further reducing the terms to their simplest form by canceling it out.

Canceling Common Factors

Canceling common factors is crucial when simplifying rational expressions. After successfully factoring the numerator and the denominator, look for identical factors that appear in both, because these can be canceled to simplify the expression. In our exercise, both the numerator \(12xy\) and the denominator \(42y\) have shared factors:- From the numbers: We identified a common factor of 6 by factoring the coefficients 12 and 42.
Additionally, the variable \(y\) appears in both parts of the expression. Once you spot these shared factors, cancel them out:- When we cancel 6 from the numeric coefficients, and \(y\) from both the numerator and denominator, the expression simplifies progressively from \(\frac{12xy}{42y}\) to \(\frac{2x}{7}\).
By canceling these factors, we are effectively presenting the expression in its simplest possible form, making it much easier to interpret or use in subsequent calculations.