Problem 81 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$ \frac{a}{2} \text { when } a=8 $$

Step-by-Step Solution

Answer

The value of the expression when \( a = 8 \) is 4.

Step 1: Substitute the Value

The first step is to substitute the value of \( a \) into the equation. This gives: \( \frac{a}{2} = \frac{8}{2} \)

Step 2: Calculate the Result

The next step is simple division: \( \frac{8}{2} = 4 \)

Key Concepts

Substitution in Algebra

In algebra, substitution is a fundamental technique used to evaluate expressions. It involves replacing variables in an algebraic expression with their corresponding numerical values. To ensure accuracy, it's essential to maintain the original structure of the expression while performing the substitution.

For example, if an expression such as \( \frac{a}{2} \) needs to be evaluated and we know that \( a = 8 \), the first step is to replace the variable \( a \) with the number 8. The resulting expression would be \( \frac{8}{2} \) after the substitution. This step doesn't involve any calculations yet—it's simply a direct replacement that sets up the expression for further arithmetic operations.

Arithmetic Division

Once substitution has been carried out, arithmetic operations come into play. Division is one of the basic arithmetic operations where we find how many times one number is contained within another. It’s essentially breaking down a larger number into equal parts.

In our example, after substituting 8 for \( a \) in \( \frac{a}{2} \), we perform arithmetic division to simplify \( \frac{8}{2} \). This process involves breaking down the number 8 into two equal parts to find that each part is 4. Hence, when we divide 8 by 2, the quotient is 4. Division is a crucial step in solving algebraic expressions when variables are substituted with numbers.

Algebraic Expression Evaluation

The evaluation of an algebraic expression is the process of finding its numerical value given the values of the variables. Once you've substituted the variables with their actual values, as shown in the previous steps, you carry out the necessary arithmetic operations to evaluate the expression.

It's important to follow the order of operations—often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)—to ensure correct evaluation.

Exercise Improvement Advice

• Always check for common mistakes such as not properly substituting all instances of the variable or ignoring the order of operations.
• Double-check your arithmetic calculations.
• Use parentheses to clarify substitution and enforce the order of operations where necessary.

When you've carefully followed these steps, you can confidently arrive at a solution. In the given exercise, the correct evaluation of \( \frac{a}{2} \) when \( a = 8 \) is 4.