Problem 19 - Elementary and Intermediate Algebra

Question

The Try Exercises for examples are indicated by a shaded block on the exercise number. Answers to these exercises appear at the end of the exercise set as well as at the back of the book. $$ \frac{m-n}{2}, \text { for } m=20 \text { and } n=6 $$

Step-by-Step Solution

Answer

7

Step 1: Substitute the values of m and n

In the expression \(\frac{m - n}{2}\), substitute \(m = 20\) and \(n = 6\). The expression then becomes \(\frac{20 - 6}{2}\).

Step 2: Simplify the numerator

Calculate the value inside the numerator: \(20 - 6 = 14\). So, the expression now is \(\frac{14}{2}\).

Step 3: Perform the division

Divide the numerator by the denominator: \(14 \div 2 = 7\).

Step 4: Final Answer

The value of the expression is \(7\).

Key Concepts

Substitution in Algebra

Substitution in algebra is an important concept that involves replacing variables in an expression with their given values. This technique is extremely useful when you are given specific values for variables and need to evaluate an expression.

For example, consider the expression \( \frac{m-n}{2} \); to evaluate it, you substitute the values provided for \( m \) and \( n \). In our case, \( m = 20 \) and \( n = 6 \). This changes the expression to \( \frac{20 - 6}{2} \).

Substitution helps simplify expressions, making them easier to solve. It’s essentially plugging in the values where the variables are and then performing arithmetic operations.

When substituting values, it's crucial to ensure that each variable is swapped out correctly to avoid calculation errors. Double-checking your substituted values is always a good idea!

Numerator and Denominator

The concepts of numerator and denominator are fundamental in understanding fractions and rational expressions. In any fraction \( \frac{a}{b} \), the numerator is the number above the line (\( a \)), and the denominator is the number below the line (\( b \))

For the given expression, \( \frac{20-6}{2} \):

- **Numerator**: The part above the fraction line, which is \( 20 - 6 \)
- **Denominator**: The part below the fraction line, which is \( 2 \)

Simplifying the numerator involves performing the operation given, which in this case is subtraction: \( 20 - 6 = 14 \). The simplified expression now is \( \frac{14}{2} \).

Understanding the role of the numerator and denominator helps in carrying out operations with fractions, simplifying them, and performing divisions correctly. Always keep an eye on simplifying both the numerator and the denominator to ensure accurate results.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These are the foundational tools that help in evaluating expressions.

In the expression given \( \frac{m-n}{2} \), we applied:

- **Subtraction**: To the numerator, by calculating \( 20 - 6 = 14 \)
- **Division**: To the simplified expression \( 14 / 2 \)

Here’s how each operation works:

- **Addition (+)**: Combines values into a larger number
- **Subtraction (-)**: Takes away one number from another
- **Multiplication (×)**: Repeated addition of a number
- **Division (÷)**: Splits a number into equal parts

Executing these operations step-by-step ensures the final answer is accurate. For our example, performing the division finally took us from \( \frac{14}{2} \) to the final answer, \( 7 \). This step-by-step breakdown helps in avoiding mistakes and understanding the flow of calculations.

Mastery of these basic operations is key to solving more complex math problems effortlessly.