Problem 17 - 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. $$ \frac{x+y}{4}, \text { for } x=2 \text { and } y=14 $$

Step-by-Step Solution

Answer

4

Step 1 - Substitute the Values

First, take the given values for the variables. Here, we have \(x = 2\) and \(y = 14\). Substitute these values into the expression \( \frac{x + y}{4} \).

Step 2 - Add the Variables

Add the substituted values of the variables: \(2 + 14\). This results in \(16\).

Step 3 - Divide by 4

Finally, divide the sum by 4: \( \frac{16}{4} \). This gives us the result \(4\).

Key Concepts

Substitution

Substitution in algebra involves replacing variables with their given numerical values. It is a fundamental concept in simplifying and solving algebraic expressions. Let's break down the process even further. Say you have an expression like \( \frac{x + y}{4} \) and values given as \( x = 2 \) and \( y = 14 \). To apply substitution, you simply replace \( x \) with 2 and \( y\) with 14. After substitution, the expression becomes \( \frac{2 + 14}{4} \). Remember, substitution helps you move from the abstract (letters) to the concrete (numbers), making it easier to understand and simplify the expression.

Addition

Addition is one of the basic arithmetic operations and is essential in algebra. When you substitute variables with their values, the next step is often to perform addition or another arithmetic operation. For example, after substituting \( x = 2 \) and \( y = 14 \) into \( \frac{x + y}{4} \), you get \( \frac{2 + 14}{4} \). Before you divide by 4, you need to add 2 and 14. Performing the addition, \( 2 + 14 \) results in 16, simplifying your expression to \( \frac{16}{4} \). Addition helps in bringing together individual values to get a single, simplified number that makes further calculations easier.

Division

Division is another key arithmetic operation crucial in algebra. After substituting and adding, you often need to divide to further simplify the expression. In our example, we end up with \( \frac{16}{4} \). Division means splitting the number 16 into 4 equal parts. Performing the division, \( \frac{16}{4} \), results in 4. This is your final answer. Understanding division is vital because it frequently appears in algebraic expressions and equations, whether you're working with simple numbers or complex polynomials.