Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression when \(x=3, y=-2\), and \(z=4$$\frac{x-y}{5 z}\)

Step-by-Step Solution

Answer

The evaluated expression is 0.25

Step 1: Substitute the variables

First, substitute the provided values into the expression. This gives: \(\frac{3-(-2)}{5*4}\)

Step 2: Perform the operations in the numerator

Subtract -2 from 3 in the numerator, which results in: \(\frac{3+2}{5*4}\)

Step 3: Simplify the numerator

Simplify the addition in the numerator, which results in: \(\frac{5}{5*4}\)

Step 4: Perform the operation in the denominator

Multiply 5 by 4 in the denominator, resulting in: \(\frac{5}{20}\)

Step 5: Simplify the expression

Simplifying the fraction \(\frac{5}{20}\) gives the final answer of \(0.25\)

Key Concepts

Substitution

When evaluating expressions, substitution is a key step. It involves replacing variables with provided values. For example, in the expression \(\frac{x-y}{5z}\), we substitute \(x\), \(y\), and \(z\) with their given values: 3, -2, and 4 respectively. This transforms our expression into \(\frac{3-(-2)}{5 \times 4}\).
Substitution makes an abstract expression concrete, allowing us to perform calculations. It is crucial to do this accurately, ensuring each variable is replaced with its correct corresponding value. Pay attention to signs, especially when dealing with negative numbers like \(-2\). Proper substitution lays the groundwork for further steps, like simplifying.

Numerator and Denominator

The terms numerator and denominator are fundamental in understanding fractions. In a fraction \(\frac{a}{b}\), \(a\) is the numerator, located above the fraction line, and \(b\) is the denominator, which is positioned below the line.

• The numerator represents how many parts of the whole are being considered.
• The denominator shows how many equal parts the whole is divided into.

In our exercise, after substitution, we have \(\frac{5}{20}\). Here, 5 is the result of subtracting \(-2\) from 3, and it becomes the numerator. The multiplication of \(5 \times 4\) gives us 20 in the denominator. Keep track of these operations separately to avoid confusion.

Simplifying Fractions

Now comes simplifying fractions, a process made to make expressions more digestible. Simplification aims to write the fraction in its simplest form, where the numerator and denominator share no common divisors besides 1.
To simplify \(\frac{5}{20}\):

• Identify the greatest common divisor (GCD) of 5 and 20, which is 5.
• Divide both the numerator and denominator by the GCD.
• This gives \(\frac{5 \div 5}{20 \div 5} = \frac{1}{4}\).

As a decimal, this fraction is equivalent to 0.25. Simplifying helps because it presents the solution in the easiest, most understandable way.