Problem 81 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{3 x+6 y}{x+2 y} $$

Step-by-Step Solution

Answer

The expression simplifies to 3.

Step 1: Identify Common Factors

Look at the numerator, \(3x + 6y\), and the denominator, \(x + 2y\). Check for common factors. In the numerator, you can factor out a 3, giving \(3(x + 2y)\).

Step 2: Simplify the Expression

After factoring, the expression is \(\frac{3(x+2y)}{x+2y}\). Identify that \(x + 2y\) is a common factor in both the numerator and the denominator.

Step 3: Cancel Out the Common Factors

Cancel the \(x + 2y\) term in the numerator and the denominator, which leaves \(3\).

Step 4: Write the Simplified Expression

The expression simplifies to \(3\). The simplified form of the given expression is \(3\).

Key Concepts

Factoring Algebraic Expressions

Factoring algebraic expressions involves breaking down expressions into their simplest components. This process is essential in simplifying rational expressions. When you have an algebraic expression, like in our original problem, the goal is to express it in terms of its factors. Let's take a closer look at this process.

Understanding factors is the first step. Factors are numbers or expressions that can be multiplied together to get the original number or expression. For example, the factors of 12 are 3 and 4 because 3 multiplied by 4 gives 12. In algebra, the same applies to expressions. For the expression in our exercise, the numerator was factored as follows:

• The original expression is: \(3x + 6y\)
• Identify common numerical and variable factors: In this case, 3 can be factored out from both terms, simplifying it to \(3(x + 2y)\).

Once factored, it becomes easier to simplify the expression by spotting common factors that can be cancelled, which leads us to our next core concept.

Common Factors

Common factors play a crucial role in simplifying rational expressions. A common factor is a term that appears in both the numerator and the denominator of a fraction. By identifying these common factors, you can simplify the expression significantly.

In our exercise, after factoring the numerator \(3x + 6y\) to \(3(x + 2y)\), we noticed that both the numerator \(3(x + 2y)\) and the denominator \(x + 2y\) shared a common factor: \(x + 2y\).

• Look at both parts: Identify the repeating term in the numerator and the denominator.
• By recognizing shared components, these terms can be canceled out from both the numerator and the denominator.
• This leaves a simplified expression: In this case, the expression simplifies to just the constant \(3\).

Understanding common factors allows for the reduction of complexity in algebraic expressions, leading to their simplest forms.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its most compact and simple form without changing its value.

The primary aim is to manage and solve expressions efficiently. Simplification involves reducing the number of terms in an expression, canceling common factors, and unifying like terms.

For the expression \(\frac{3x + 6y}{x + 2y}\), simplification followed these steps:

• Factor the numerator: Convert \(3x + 6y\) to \(3(x + 2y)\).
• Identify common factors: Recognize \(x + 2y\) as a common factor in the fraction.
• Cancel out these common terms: Removing \(x + 2y\) from both the top and bottom leads to just \(3\).

Simplification helps in making complex expressions manageable and comprehensible, streamlining calculations and solving algebraic tasks more easily.