Problem 24 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ \frac{d}{12} \text { when } d=36 $$

Step-by-Step Solution

Answer

The evaluated expression when \(x = 7\) is 47.

Step 1: Substitute the value of x

First, substitute \(x = 7\) into the expression. The expression \(x^{2}-2\) becomes \(7^{2}-2\).

Step 2: Simplify squared term

Square 7 to get 49. So, the expression becomes 49 - 2.

Step 3: Perform the subtraction

Subtract 2 from 49 to get the final answer. So, 49 - 2 equals 47.

Key Concepts

Substitution in Algebra

Substitution is an essential process in algebra that involves replacing variables with their corresponding numerical values. This method is used extensively when evaluating algebraic expressions. Let's take a practical approach to understand it better. Imagine an algebraic expression as a recipe, and the variables are like ingredients that we need to replace with actual food items to get the final dish.

For example, if we have an expression like x^{2} - 2 and we're told that x=7, substitution means we replace every instance of x in our expression with 7. Think of it as swapping out a placeholder for its real value. After substitution, our expression becomes 7^{2} - 2, which is the first critical step towards finding our answer.

Simplifying Algebraic Expressions

Simplifying an algebraic expression is a process of making it as straightforward and concise as possible. This normally involves performing operations like addition, subtraction, multiplication, and division, as well as applying the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

Using our previous example, after substituting x with 7 to get 7^{2} - 2, the next step is to simplify the squared term. To simplify 7^{2}, we multiply 7 by itself to get 49. Our expression now looks much simpler: 49 - 2. Simplification is all about making the expression easier to understand and solve, just as organizing a pantry makes it easier to find ingredients for cooking.

Performing Arithmetic Operations

Arithmetic operations are the building blocks of mathematics. They include addition, subtraction, multiplication, and division. When performing arithmetic operations in algebra, it's crucial to follow the correct order to avoid mistakes. In our example expression, 49 - 2, there is only one operation left to perform: subtraction.

To determine the final answer, we simply take away 2 from 49, which gives us 47. This step seals the deal on our evaluation of the algebraic expression. It's much like finishing a puzzle—you have all the right pieces, and now it's just about putting them in their place to see the complete picture.