Problem 62 - Algebra A Combined Function

Question

Evaluate each expression when \(x=12, y=8,\) and \(z=4\). $$ \frac{y}{z}+8 x $$

Step-by-Step Solution

Answer

The value of the expression is 98.

Step 1: Substitute the given values

First, substitute the values for each variable given in the problem. Replace \(y\) with 8, \(z\) with 4, and \(x\) with 12 in the expression \(\frac{y}{z} + 8x\). Thus, the expression becomes \(\frac{8}{4} + 8 \cdot 12\).

Step 2: Perform division

Calculate the division in the expression. Divide 8 by 4 to get the first part of the expression. Thus, \(\frac{8}{4} = 2\). Now the expression simplifies to \(2 + 8 \cdot 12\).

Step 3: Multiply values

Now, perform the multiplication. Multiply 8 by 12 to obtain the second part of the expression. Calculating, \(8 \cdot 12 = 96\). Therefore, the expression is now \(2 + 96\).

Step 4: Add results

Finally, add the results of the previous calculations. Add 2 and 96 together to obtain the total result. Calculating, \(2 + 96 = 98\). Therefore, the final result of the original expression is 98.

Key Concepts

Variable Substitution

Variable substitution is a fundamental concept in algebra. It's like filling in the blanks with specific numbers. Whenever you see a variable in an expression, you can replace it with a given value. In this exercise, variable substitution involves the following steps:

• Identify each variable in the expression. In our example, the variables are \( x \), \( y \), and \( z \).
• Replace each variable with its given value. For instance, substitute \( y=8 \), \( z=4 \), and \( x=12 \) into the expression \( \frac{y}{z} + 8x \).
• The expression now reads \( \frac{8}{4} + 8 \cdot 12 \).

Substitution is crucial because it turns an abstract algebraic expression into a more tangible arithmetic problem that you can solve step-by-step.

Order of Operations

To solve any mathematical expression accurately, you must follow the correct order of operations. The sequence typically follows the **PEMDAS/BODMAS** rule:

• **P/B**: Parentheses/Brackets first
• **E/O**: Exponents/Orders (like squares and square roots) next
• **MD**: Multiplication and Division from left to right
• **AS**: Addition and Subtraction from left to right

In this exercise, after substituting the variables, the expression becomes \( \frac{8}{4} + 8 \times 12 \). According to the order of operations:

• Perform the division \( \frac{8}{4} \) before anything else, which simplifies to 2.
• Next, calculate the multiplication \( 8 \times 12 \).
• Finally, handle the addition of the result of \(2 + 96\).

Following these steps ensures that you arrive at the correct solution without errors.

Mathematical Evaluation

Once you've substituted variables and followed the proper order, it's time for mathematical evaluation. This means you actually compute the value of the expression.The final step-by-step computation includes:

• Calculate \( \frac{8}{4} = 2 \) as the division outcome.
• Multiply \( 8 \times 12 = 96 \) as the result of the multiplication step.
• Add the outcomes \( 2 + 96 = 98 \) together.

Mathematical evaluation is about doing these operations accurately to reach the final answer. Thus, for the given expression \( \frac{y}{z} + 8x \) with substituted values, the final output is 98.Each step plays a vital role, allowing you to break down complex expressions into manageable arithmetic parts. Always double-check your steps to ensure the accuracy of your results.