Problem 10 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{21 x y}{35 x}\)

Step-by-Step Solution

Answer

Simplified expression is \( \frac{3y}{5} \).

Step 1: Identify Common Factors

First, let's identify the common factors in the numerator and the denominator. Numerator: 21xy Denominator: 35x We notice that both 21 and 35 are divisible by 7, and both terms contain at least one x.

Step 2: Simplify the Coefficient

We divide both 21 and 35 by their greatest common divisor, which is 7: 21 ÷ 7 = 3 35 ÷ 7 = 5 So now the expression looks like: \[ \frac{3xy}{5x} \]

Step 3: Simplify the Variable x

Since the variable x appears in both the numerator and denominator, we can cancel one x from each, leaving us with: \[ \frac{3y}{5}\]

Step 4: Write the Final Simplified Expression

Combine the simplified coefficient and variable terms to get the final result:\[ \frac{3y}{5} \]

Key Concepts

Common Factors

Understanding common factors is crucial when simplifying rational expressions. A common factor is a number or variable that divides exactly into two or more numbers or expressions. For example, in the fraction \( \frac{21xy}{35x} \), we need to find common factors in the numerator and denominator.

• The number 21 can be broken down into factors of 3, 7, and \( x \).
• The number 35 can be broken down into factors of 5 and 7, with a variable \( x \).

Here, both 21 and 35 share a common factor of 7, and both terms contain the factor \( x \). Identifying these common factors allows us to simplify rational expressions effectively.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It's integral in reducing fractions to their simplest form. In our example of \( \frac{21xy}{35x} \), we determined that both 21 and 35 are divisible by 7. This number, 7, is their greatest common divisor.

• Divide the numerator 21 by 7 to get 3.
• Divide the denominator 35 by 7 to get 5.

This division adjusts the coefficients and results in a simpler fraction: \( \frac{3xy}{5x} \). Using the GCD in this way streamlines the simplification process.

Cancelling Terms

Cancelling terms in a rational expression involves eliminating common factors from both the numerator and denominator. This makes the expression simpler and easier to understand. For \( \frac{3xy}{5x} \), we notice the variable \( x \) appears in both parts of the expression.

• By cancelling out the common factor \( x \) in the numerator and denominator, we effectively reduce the complexity.
• We are left with \( \frac{3y}{5} \), a much simpler expression.
  

This cancellation process is legitimate because dividing a number by itself yields one, and multiplying by one does not change the value of an expression.

Numerator and Denominator

The numerator and denominator are the top and bottom parts of a fraction, respectively. In simplifying rational expressions, like \( \frac{21xy}{35x} \), it's essential to examine and manipulate both parts separately to get a simplified result.

• The numerator of a fraction provides the count of parts you have. In our example, it starts as \( 21xy \).
• The denominator indicates how many equal parts the whole is divided into, starting as \( 35x \).

By identifying and cancelling common factors between these two components, we transition from a complex fraction to \( \frac{3y}{5} \), where the counting and partitioning are clear and simplified. This process clarifies the roles of the numerator and denominator in fraction manipulation.