Problem 11 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(x=6 .\) See Example 1. $$ \frac{x-2}{x-5} $$

Step-by-Step Solution

Answer

The evaluated expression is 4.

Step 1: Substitute the Value

Substitute the given value of \( x = 6 \) into the expression: \( \frac{x-2}{x-5} \). This becomes \( \frac{6-2}{6-5} \).

Step 2: Evaluate the Numerator

Calculate the numerator by performing the subtraction: \( 6 - 2 = 4 \).

Step 3: Evaluate the Denominator

Calculate the denominator by performing the subtraction: \( 6 - 5 = 1 \).

Step 4: Calculate the Result

Substitute the evaluated numerator and denominator back into the expression to get \( \frac{4}{1} \). Simplify to find the result: \( 4 \).

Key Concepts

Substitution

In algebra, substitution is a core technique where we replace a variable with a given value. This is particularly useful in evaluating algebraic expressions. Let's say your expression includes a variable. To solve it, you simply substitute the variable with the specified number provided in the exercise.
For example, if you're tasked to evaluate the expression \( \frac{x-2}{x-5} \) for \( x = 6 \), you replace every instance of \( x \) in the expression with 6:

• Original expression: \( \frac{x-2}{x-5} \)
• Substitute \( x = 6 \): \( \frac{6-2}{6-5} \)

By doing this, you transform an algebraic expression into a straightforward arithmetic calculation. This substitution serves as the first step in evaluating the expression.

Numerator and Denominator

When dealing with fractions, understanding the terms numerator and denominator is essential. In any fraction, the term above the line is called the numerator, and the term below is the denominator. Together, they dictate the value of the fraction.
For the expression \( \frac{x-2}{x-5} \), after substitution with \( x = 6 \), we have:\

• Numerator: \( 6-2 \), which simplifies to 4.
• Denominator: \( 6-5 \), which simplifies to 1.

The numerator indicates how many parts of the whole we are considering, and the denominator indicates how many equal parts the whole is divided into. Properly calculating both these components bridges us to the simplification of the expression.

Simplification

Simplification involves reducing an expression to its most basic form. Once you have calculated the numerator and denominator, you can simplify the fraction by dividing the numerator by the denominator. This gives us the simplest form of the expression.
In \( \frac{4}{1} \), the 4 can be divided by 1, since any number divided by 1 remains unchanged, simplifying nicely to 4. This simplification process can be crucial for understanding and solving more complex algebraic expressions.
Key points to remember during simplification:

• Always simplify fractions by finding the greatest common factor, if possible.
• If the denominator is 1, the fraction simplifies directly to the numerator.

Through these steps, simplification streamlines the expression, leading to an easy-to-understand result.