Problem 20 - Prealgebra

Question

Evaluate each expression if \(x=7, y=3,\) and \(z=9\) $$4 z-3 y$$

Step-by-Step Solution

Answer

The value of the expression is 27.

Step 1: Substitute Values

First, substitute the given values of the variables into the expression. The expression is \(4z - 3y\). Given \(z = 9\) and \(y = 3\), substitute these values to get \(4(9) - 3(3)\).

Step 2: Perform Multiplications

Next, perform the multiplications in the expression. Calculate \(4 \times 9 = 36\) and \(3 \times 3 = 9\). The expression now becomes \(36 - 9\).

Step 3: Calculate the Difference

Finally, perform the subtraction to find the value of the expression: \(36 - 9 = 27\).

Key Concepts

Evaluating Expressions

Evaluating expressions is all about finding the value of an algebraic expression by substituting given numerical values for each variable in the expression. This process requires precision and attention to detail.

• Substitute the Values: Start by identifying the variables in the expression and replace them with the given numeric values. For instance, in the expression \(4z - 3y\), replace \(z\) with 9 and \(y\) with 3 to get \(4(9) - 3(3)\).
• Follow the Steps: It's crucial to perform this substitution step carefully, ensuring you input the correct values for each variable.
• Checking Work: After substituting, it's a good practice to re-evaluate your substitutions to make certain no errors have been made before moving forward.

Order of Operations

Order of operations is a fundamental concept in math that dictates the sequence in which operations should be carried out to correctly evaluate an expression. A popular acronym to remember this sequence is PEMDAS:

• P: Parentheses come first.
• E: Exponents (i.e., powers and roots, etc.) follow next.
• M/D: Multiplication and Division (left to right).
• A/S: Addition and Subtraction (left to right).

Applying this to our expression \(4(9) - 3(3)\):

• We handle multiplication before subtraction based on PEMDAS. Calculate \(4 \times 9 = 36\) and \(3 \times 3 = 9\). Therefore, our expression becomes \(36 - 9\).
• Next, perform the subtraction: \(36 - 9 = 27\).

Following the correct order of operations is crucial to reaching the right answer.

Arithmetic Operations

Arithmetic operations are the basics of solving any mathematical expression, involving operations like addition, subtraction, multiplication, and division. Let's briefly explain the operations involved in our example:

• Multiplication: It's the process of adding a number to itself a certain number of times. For example, \(4 \times 9\) can be visualized as adding 9, four times, reaching a product of 36.
• Subtraction: This operation involves finding the difference between two numbers. In our computation, we had to subtract 9 from 36, yielding 27.

Understanding these operations thoroughly ensures you can correctly manipulate and solve expressions in algebra.