Problem 49 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5.\) \(\frac{z}{5 x}\)

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Identify the variables and their values

We are given the expression \( \frac{z}{5x} \) and the values for the variables: \( x = 1 \), \( y = 3 \), and \( z = 5 \). Note that \( y \) is not needed for this problem.

Step 2: Substitute the values into the expression

Replace \( x \) and \( z \) with their given values in the expression \( \frac{z}{5x} \). This gives \( \frac{5}{5 \times 1} \).

Step 3: Simplify the expression

Calculate the expression \( \frac{5}{5 \times 1} \). First, compute the denominator: \( 5 \times 1 = 5 \). Then divide 5 by 5: \( \frac{5}{5} = 1 \).

Key Concepts

Variable Substitution

Variable substitution is a fundamental concept in algebra where we replace variables with specific values. In this exercise, the expression given is \( \frac{z}{5x} \). We have values for the variables: \( x = 1 \) and \( z = 5 \). When the problem provides specific numbers for variables, the first step is to carefully replace each variable with its assigned value.
Here, you substitute \( z \) with 5 and \( x \) with 1. This changes the expression from \( \frac{z}{5x} \) to \( \frac{5}{5 \times 1} \). Make sure that each variable is substituted correctly to avoid any errors in further calculations.
Tip for success: Pay close attention to ensure that only the necessary variables are replaced, like in this case, where \( y = 3 \) is irrelevant and is therefore ignored. This helps streamline your work and keeps it focused.

Expression Evaluation

Expression evaluation involves calculating the value of an expression after substituting variables with their respective values. Once variables are substituted, the next step is to evaluate the calculated numerical expression.
In our example with \( \frac{5}{5 \times 1} \), we begin by addressing the operation in the denominator, which is \( 5 \times 1 \). It's important to operate following the mathematical order of operations—often denoted as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Once you solve \( 5 \times 1 = 5 \), the expression to evaluate becomes \( \frac{5}{5} \). This step is crucial as it determines the final numerical result of the expression.

Simplification of Expressions

Simplification of expressions is the final step where you reduce the expression to its simplest form. After evaluating the expression \( \frac{5}{5} \), the aim is to simplify it further if possible.
In this case, we can simplify \( \frac{5}{5} \) by dividing the numerator by the denominator. Since 5 divided by 5 equals 1, the simplest form of this expression is 1.
Why is simplification important? It provides a clearer view of the expression's value and often makes it easier to understand or compare with other mathematical expressions. Being able to simplify effectively also enhances your problem-solving efficiency and mathematical fluency.