Problem 19 - Prealgebra

Question

Evaluate each expression if \(x=7, y=3,\) and \(z=9\) $$3 x-2 y$$

Step-by-Step Solution

Answer

The value of the expression is 15.

Step 1: Understand the Expression

The given expression is \(3x - 2y\). We need to evaluate this expression by substituting the given values of \(x\) and \(y\).

Step 2: Substitute the Values

Insert \(x=7\) and \(y=3\) into the expression:\[ 3(7) - 2(3) \]

Step 3: Calculate the Products

Calculate the products in the expression:- Compute \(3 \times 7 = 21\)- Compute \(2 \times 3 = 6\)

Step 4: Perform the Subtraction

Subtract the second product from the first:\[ 21 - 6 = 15 \]

Step 5: Conclusion

The value of the expression \(3x - 2y\), when \(x=7\) and \(y=3\), is \(15\).

Key Concepts

Substitution

The concept of substitution is all about replacing variables in an algebraic expression with actual numbers. Think of variables as placeholders in the expression. In mathematical problems, these placeholders are usually given specific values. Substitution is the process where we replace these placeholders with the numbers they stand for.
For example, let's take the expression given in the problem: \(3x - 2y\). Here, \(x\) and \(y\) are variables. When we're told that \(x = 7\) and \(y = 3\), substitution involves replacing \(x\) with 7 and \(y\) with 3.
So, to perform substitution in this problem, we rewrite the expression with these values:

• Replace \(x\) with 7 to get \(3(7)\).
• Replace \(y\) with 3 to get \(2(3)\).

This step prepares the expression for further arithmetic operations.

Evaluation

Evaluation involves calculating the numerical value of an expression after substitution. Once we've substituted in the specific numbers for each variable, we then perform the arithmetic operations to solve the expression.
With the expression \(3x - 2y\), and after substituting \(x = 7\) and \(y = 3\), we're left with the expression \(3(7) - 2(3)\).
Evaluation requires us to:

• First compute \(3 \times 7\), which equals 21.
• Next calculate \(2 \times 3\), which results in 6.

After computing these products, the expression is simplified so we can proceed to the final step of evaluation: performing the arithmetic to get the final result.

Basic Arithmetic Operations

Arithmetic operations form the backbone of evaluating algebraic expressions. In this context, the main operations are multiplication and subtraction. Understanding and executing these basic operations is crucial to solving expressions correctly.
In our specific example from the problem \(3(7) - 2(3)\):

• Multiplication: This operation is used to calculate \(3 \times 7 = 21\) and \(2 \times 3 = 6\).
• Subtraction: After obtaining the products, we perform subtraction. Here, subtract 6 from 21 to find the result: \(21 - 6 = 15\).

These operations collectively allow us to find the numerical value of the expression and arrive at the conclusion that \(3x - 2y = 15\) when \(x = 7\) and \(y = 3\). By mastering these basic arithmetic operations, you can efficiently evaluate any algebraic expression.