Problem 109 - Elementary Algebra

Question

Explain how you would simplify \(\frac{12 x^{2} y}{18 x y}\).

Step-by-Step Solution

Answer

The simplified form is \(\frac{2x}{3}\).

Step 1: Factor the Numerator and Denominator

We start by factoring both the numerator and the denominator of the given fraction. The numerator is \(12x^2y\), which can be factored as \(2 \times 2 \times 3 \times x \times x \times y\). The denominator is \(18xy\), which can be factored as \(2 \times 3 \times 3 \times x \times y\).

Step 2: Identify and Cancel Common Factors

Next, identify the common factors in both the numerator and the denominator. Both have \(2\), \(3\), \(x\), and \(y\) as factors. Cancel these common factors from both the numerator and denominator. After canceling, the numerator becomes \(2x\) and the denominator becomes \(3\).

Step 3: Write the Simplified Fraction

After canceling the common factors, write the remaining terms of the fraction. This gives us the simplified fraction \(\frac{2x}{3}\).

Key Concepts

Factoring Numerators and Denominators

When simplifying fractions, one of the first steps is to break down both the numerator and the denominator into their prime factors. This process is called 'factoring.' It helps us see all the components that make up the numbers involved.
For instance, if we have the fraction \(\frac{12x^2y}{18xy}\), we start by factoring each part:

• The numerator \(12x^2y\) can be broken into its components: \(2 \times 2 \times 3 \times x \times x \times y\).
• The denominator \(18xy\) can be broken down to: \(2 \times 3 \times 3 \times x \times y\).

When you factor, you're essentially reducing each term to its simplest building blocks. This lets you clearly see the common components between the numerator and the denominator, which is crucial for the next step. Factoring can also help in recognizing patterns or simplifying more complex algebraic expressions.

Canceling Common Factors

After you've factored both the numerator and the denominator, it's time to simplify the fraction by canceling out any common factors.
This means removing the factors that appear in both the numerator and the denominator. In our example, both \(12x^2y\) and \(18xy\) share the factors \(2\), \(3\), \(x\), and \(y\).

• We cancel one \(2\) from both the numerator and denominator.
• Next, we cancel one \(3\) from each.
• We also cancel one \(x\) and one \(y\) from both.

After canceling out these common factors, you're left with the 'leftover' parts of the fraction. In this example, you're left with \(\frac{2x}{3}\). Canceling common factors doesn't change the value of the fraction; it just simplifies it. This step helps to reduce complexity in solving or understanding algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols. Simplifying them is essential for problem-solving in math because it makes expressions easier to work with.
Consider the expression \(\frac{12x^2y}{18xy}\). Here, \(x\) and \(y\) are variables, representing unknown or generic values. The goal is to get the simplest version of this expression without losing any information about the relationships between variables.

• Factoring helps by breaking expressions into simpler parts.
• Canceling common factors reduces complexity.

By executing these steps, you transform complex algebraic expressions into much simpler forms. This makes it much easier to use them in equations, compare them, or apply them to real-world situations. Ultimately, simplifying algebraic expressions is a skill that, once mastered, paves the way for understanding more advanced mathematical concepts.