Problem 13 - Introductory Algebra for College Students

Question

Evaluate each expression for \(x=4\). $$\frac{12 x-8}{2 x}$$

Step-by-Step Solution

Answer

The evaluated expression equals 5

Step 1: Substitution

Substitute \(x = 4\) into the algebraic expression. That gives us\[\frac{12*4 - 8}{2*4}\]

Step 2: Calculate Numerator and Denominator Separately

Perform the operations in the numerator and the denominator separately. In the numerator, \(12*4 - 8\) equals \(48 - 8\) which is \(40\). In the denominator, \(2*4\) is equal to \(8\). So the expression becomes\[\frac{40}{8}\]

Step 3: Simplify the Fraction

Finally, simplify the fraction \(\frac{40}{8}\) by dividing the numerator and the denominator by their greatest common divisor, which is \(8\). So,\[\frac{40}{8} = 5\]

Key Concepts

Substitution Method

The substitution method is a straightforward technique used to evaluate algebraic expressions. When given an expression like \( \frac{12x-8}{2x} \), and you need to find the value at a specific point, you simply replace each occurrence of the variable in the expression with the given number.

In this exercise, we are asked to evaluate the expression for \( x=4 \). To substitute, take each \( x \) in the expression \( \frac{12x-8}{2x} \) and replace it with \( 4 \). This transforms the expression into:

• \( 12 \times 4 - 8 \)
• \( 2 \times 4 \)

This substitution is the first crucial step in evaluating expressions, setting up the foundation for further simplification.

From here, you can proceed with solving each part separately to move on to the calculations of the numerator and denominator.

Simplifying Fractions

Simplifying fractions is an important skill in mathematics that makes expressions easier to understand and work with. After performing operations separately in the numerator and the denominator, as we've done in this exercise, we arrive at the fraction \( \frac{40}{8} \).

The goal of simplifying is to reduce the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this situation, both 40 and 8 are divisible by 8:

• 20 divided by 8 gives 5.
• 8 divided by 8 gives 1.

So, \( \frac{40}{8} \) simplifies to \( 5 \). Simplifying fractions helps reveal their most basic form, making them more practical for use in larger calculations or expressions.

Numerator and Denominator Calculation

Handling numerators and denominators separately can vastly simplify evaluation of arithmetic expressions. The given fraction \( \frac{12x - 8}{2x} \), after substituting \( x = 4 \), turns into \( \frac{48 - 8}{8} \).

First, calculate the numerator:

• Multiply 12 by 4 to get 48.
• Subtract 8 from 48 to yield 40.

Next, compute the denominator:

• Multiply 2 by 4, resulting in 8.

This methodical approach ensures that each arithmetic operation is correctly performed, leading to the reduced fraction: \( \frac{40}{8} \). By splitting tasks into smaller parts, fractions are easier to handle, reducing errors and simplifying the end result.