Problem 20 - 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. $$ \frac{x-y}{6}, \text { for } x=23 \text { and } y=5 $$

Step-by-Step Solution

Answer

3

Step 1: Identify the Given Values

Extract the given values for the variables from the problem statement: For this exercise, we have: - \( x = 23 \) - \( y = 5 \)

Step 2: Substitute Values into the Expression

Substitute the given values of \( x \) and \( y \) into the expression \( \frac{x-y}{6} \): \[ \frac{23 - 5}{6} \]

Step 3: Simplify the Numerator

First, simplify the numerator by performing the subtraction: \[ 23 - 5 = 18 \]

Step 4: Divide by the Denominator

Next, divide the result from the numerator by the denominator of 6: \[ \frac{18}{6} = 3 \]

Step 5: Verify the Solution

Finally, verify the arithmetic to ensure accuracy: Since \( 18 \div 6 = 3 \), the 'Verify the Solution' step is completed.

Key Concepts

substitution in algebra

In algebra, substitution is a fundamental concept used when solving equations or expressions. It involves replacing the variables in an expression with their given numerical values. For example, in the exercise \frac{x-y}{6}, we substitute the given values: \[ x = 23 \ y = 5 \], into the expression, yielding: \[ \frac{23 - 5}{6} \]. This helps to simplify the problem and makes calculation easier. Substitution ensures that you're working with concrete numbers rather than abstract variables, which is especially helpful in verifying your solutions.

simplifying expressions

Simplifying an expression means to perform all possible arithmetic operations to reduce it to its simplest form. After substituting the values in our exercise, \[ \frac{23 - 5}{6} \], the next step is to simplify the numerator, which is the part of the fraction above the line. By performing the subtraction: \[ 23 - 5 = 18 \]. Now, the fraction is \[ \frac{18}{6} \], which is simpler and easier to work with than the original. Simplifying expressions is a critical algebraic skill, as it also helps in making more complex expressions easier to manage and understand.

division and subtraction

Division and subtraction are key arithmetic operations that you often encounter in algebra. In our exercise, after simplifying the numerator to 18, we then divide this result by the denominator, 6. The division process is straightforward: \[ \frac{18}{6} = 3 \]. Subtraction was used earlier to simplify the numerator: \[ 23 - 5 = 18 \]. Both operations play a significant role in reaching the final solution. Mastering these basic arithmetic skills ensures that you can tackle more complex algebraic problems with confidence.