Problem 64 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \begin{equation} \frac{z}{2} \text { when } z=8 \end{equation}

Step-by-Step Solution

Answer

The evaluation of the expression \( \frac{z}{2} \) when \( z = 8 \) equals 4.

Step 1: Identify the given expression and the value of the variable

The expression given in this problem is \( \frac{z}{2} \) and the value of the variable \( z \) is given as 8.

Step 2: Substitution of variable

Replace the variable \( z \) in the expression \( \frac{z}{2} \) with the given value of 8, obtaining \( \frac{8}{2} \).

Step 3: Perform the arithmetic operation

By simplifying \( \frac{8}{2} \), get the result as 4.

Key Concepts

Evaluating Expressions

Evaluating expressions in mathematics is the process of determining the value of an expression when the variables within the expression are replaced with given numbers. This is a fundamental concept in algebra, allowing you to find specific numerical outputs from algebraic formulas.

• Start by identifying the expression and the variables involved. In this case, the expression is \( \frac{z}{2} \) with the variable \( z \).
• Understand the given values. Here, we use \( z = 8 \).
• Substitute the given numbers for the variables in the expression to transform it from a general statement into a specific numeric equation.

In our example, we substitute \( z \) with 8 and evaluate the expression \( \frac{z}{2} \) to \( \frac{8}{2} \). This step bridges the gap between algebraic expressions and real numbers.

Arithmetic Operations

Arithmetic operations are the fundamental operations you perform when evaluating expressions. They include addition, subtraction, multiplication, and division.

In this task, you performed a division. Here's how it goes:

• Once the substitution is done, you directly handle the numeric operations. After replacing \( z \), the expression \( \frac{8}{2} \) needs calculation.
• Division breaks down into the number of times one value fits into another. Here, 8 divided by 2 is 4.
• You simplify the expression by executing the operation, resulting in a single, simplified outcome.

Performing these operations correctly is essential in algebra to move from the symbolic representation to a real-world numerical solution. Solving the expression gives you the value of the expression after the variable has been substituted.

Algebraic Expression Simplification

Simplification of algebraic expressions involves reducing expressions to their simplest form, making them easier to understand and work with.

• The primary goal is to express complex expressions in the simplest terms possible.
• In the expression \( \frac{8}{2} \), dividing 8 by 2 demonstrates simplification by reducing the fraction to its simplest integer form, which is 4.
• When you reach a single number or a simple fraction, the simplification process is complete.

This ensures that the expressions are not only easier to evaluate but also more straightforward to communicate and apply in various mathematical contexts. Effective simplification highlights the elegance of mathematics by turning multiple operations into a single, clean result.