Problem 69 - Algebra A Combined Function

Question

Perform each indicated operation and simplify. $$ \frac{3 x}{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x \).

Step 1: Identify the Operation

The operation given is a division of a single term, where \( \frac{3x}{3} \) involves dividing the product \( 3x \) by the integer 3.

Step 2: Apply Division

Apply the division operation by dividing both the numerator and the denominator by 3. This simplifies the expression: \( \frac{3x}{3} = x \).

Step 3: Verify Simplification

Check if the expression is completely simplified. Since there are no other common factors in the numerator and the denominator and only \( x \) remains, the expression is in its simplest form.

Key Concepts

Algebraic Division

Algebraic division is the process of dividing algebraic expressions. In this process, you divide the terms in the numerator by those in the denominator, which involves both numbers and variables. For our exercise, we started with the expression \( \frac{3x}{3} \). Here, the term \( 3x \) is divided by 3. Whenever you divide algebraic expressions, aim to simplify the operation:

• Identify components: Understand what you need to divide, like coefficients and terms.
• Divide systematically: Follow the arithmetic rules, starting with numbers and then variables if needed.

In this problem, the division operation simplifies to \( x \), because 3 divided by 3 gives us 1, leaving the variable \( x \) alone. It’s key to perform each division step carefully and understand what each part represents. Practice makes perfect!

Simplification Process

The simplification process is key to making algebraic expressions easier to work with. It involves reducing expressions to their simplest form by eliminating unnecessary components. For our expression \( \frac{3x}{3} \), the task was to simplify by performing division.Here's how simplification works:

• Identify like terms: Check if terms can be combined or cancelled based on common factors.
• Perform operations: Carry out addition, subtraction, multiplication, or division as needed.
• Reduce to simplest form: Eliminate coefficients that multiply out to one, where possible.

Once we divide the numerator by the denominator, we end with \( x \). This is the simplest form, as there are no more common factors to address.Remember, simplification eases solving and understanding complex algebraic problems.

Common Factors in Algebra

Common factors refer to components shared by the terms in a numerator and denominator, which can often be cancelled out to simplify expressions. Understanding common factors aids tremendously in handling algebraic expressions efficiently.In our example, consider \( \frac{3x}{3} \):

• Identify the common factor: Here, 3 is present in both the numerator and the denominator.
• Cancel common factors: Divide both components by the common factor, which simplifies the expression.

By cancelling the common factor in our expression, we simplify \( \frac{3x}{3} \) to \( x \). When you know how to identify and remove common factors, you'll find simplifying algebraic expressions much easier, leading to quicker and more accurate results in mathematical tasks.