Problem 51 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The value is 1.

Step 1: Substitute the given values

First, substitute the given values of the variables into the expression. We have \(x = 1\), \(y = 3\), and \(z = 5\). The expression is \(\frac{z}{5x}\). Replace \(z\) with 5 and \(x\) with 1.

Step 2: Simplify the expression

Now, the expression after substitution becomes \(\frac{5}{5 \times 1}\). Simplify this expression by first calculating \(5 \times 1\) which equals 5.

Step 3: Perform the division

Finally, divide 5 by 5. Since \(\frac{5}{5} = 1\), the expression simplifies to 1.

Key Concepts

Substitution in Algebra

Substitution in algebra is a fundamental concept that allows us to replace variables with their given values. When evaluating expressions, knowing how to substitute correctly is crucial. For example, in the expression \( \frac{z}{5x} \), we need the specific values of each variable to continue. Here, we were given \( x = 1 \), \( y = 3 \), and \( z = 5 \). Since the expression involves \( z \) and \( x \), we only substitute these values.
Let's break down the process:

• Identify the variables in the expression that need substitution. In our case, they are \( z \) and \( x \).
• Replace the variables with the provided values: \( z = 5 \) and \( x = 1 \).

This substitution transforms the original expression \( \frac{z}{5x} \) into \( \frac{5}{5 \times 1} \). Remember, substitution is the first step in evaluating any algebraic expression.

Simplifying Expressions

Once you've performed the substitution, the next step is to simplify the expression to find a straightforward solution. Simplifying means making the expression as simple as possible without changing its value. After substitution, our expression became \( \frac{5}{5 \times 1} \).
Here's how to simplify effectively:

• First, solve any multiplication or division operations within the expression. Here, calculate \( 5 \times 1 \) to simplify the denominator.
• The result of the multiplication is 5, so the expression simplifies further to \( \frac{5}{5} \).

Simplification helps in reducing an expression to its most basic form, making it easier to understand and solve further operations if needed.

Basic Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division are essential skills in algebra and mathematics in general. For this specific exercise, we focused on multiplication and division.
Let's see how these operations were used:

• The multiplication \( 5 \times 1 \) is straightforward. When multiplying, combine the numbers to find the result, which was 5 in this case.
• Then we perform division: the numerator 5 divided by the denominator 5 gives us \( \frac{5}{5} = 1 \).

Mastering these basic arithmetic operations enables more complex problem-solving and ensures calculations are accurate and efficient. Always remember to perform operations in the correct order and double-check your work for accuracy.