Problem 57 - Beginning and Intermediate Algebra

Question

Evaluate each expression if \(x=12, y=8,\) and \(z=4.\) \(\frac{x}{z}+3 y\)

Step-by-Step Solution

Answer

The expression evaluates to 27.

Step 1: Substitute Values

First, we need to substitute the given values of the variables into the expression. We replace \(x\) with 12, \(y\) with 8, and \(z\) with 4, so the expression becomes \(\frac{12}{4} + 3 \times 8\).

Step 2: Perform Division

Next, perform the division within the expression. Calculate \(\frac{12}{4}\), which is 3. Then, the expression simplifies to \(3 + 3 \times 8\).

Step 3: Calculate Multiplication

Now, perform the multiplication in the expression. Calculate \(3 \times 8\), which gives 24. Now the expression is simplified to \(3 + 24\).

Step 4: Add the Results

Finally, add the results from the previous step. Calculate \(3 + 24\), which equals 27.

Key Concepts

Variable Substitution

When we talk about variable substitution, it simply means replacing variables with their given numerical values. Let's imagine you have a recipe that calls for different ingredients. The ingredients are like the variables in algebra. In our exercise, we were given that \(x = 12\), \(y = 8\), and \(z = 4\). Here's how you can approach variable substitution:

• Identify the variables in the expression.
• Replace each variable with the corresponding number that you were given.

For example, in the expression \(\frac{x}{z} + 3y\), we substitute \(x\) with 12, \(y\) with 8, and \(z\) with 4, which gives us \(\frac{12}{4} + 3 \times 8\). Each variable is now a known number, making the expression easier to handle!
Substitution is a crucial part of evaluating algebraic expressions because it allows us to transform abstract variables into concrete numbers so we can calculate them.

Order of Operations

The order of operations is a set of rules that tells us the order in which to calculate an expression. This rule is essential because it ensures that everyone calculates the expressions in the same way, leading to consistent results. The order of operations is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents (or powers)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the expression \(\frac{12}{4} + 3 \times 8\), we first perform the division \(\frac{12}{4}\), which simplifies to 3. This step addresses any division, which comes before multiplication and addition in PEMDAS. Thereafter, we multiply \(3 \times 8\) to get 24, and finally, add the results of these operations. The sum of 3 and 24 gives us 27.
Remembering to apply the order of operations prevents errors and ensures the accurate evaluation of mathematical expressions.

Evaluating Expressions

Evaluating expressions is the process of calculating the value of an expression after substituting the variables with given numbers. It's like solving a puzzle where each step gets you closer to the final picture. Breaking it down to steps:

• Substitute the values of the variables (as we did earlier).
• Follow the order of operations to compute each part of the expression correctly.

For the given expression \(\frac{12}{4} + 3 \times 8\), we first made the variable substitutions. Then, respecting the order of operations, we simplified the division, performed the multiplication, and ended with the addition.
The evaluation allowed us to determine that the expression's value is 27. Evaluating expressions is an integral skill in algebra that enables us to find specific numerical results from generalized formulas or equations.