Problem 41 - Elementary Algebra Within Reach

Question

In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(3 x-2 y\) (a) \(x=4, y=3\)

Step-by-Step Solution

Answer

The evaluated result of the algebraic expression \(3x - 2y\) for the values \(x=4, y=3\) is \(6\).

Step 1: Identify the expression and the values

The algebraic expression to be evaluated is \(3x - 2y\), where \(x=4\) and \(y=3\).

Step 2: Substitute the values of the variables

Substitute \(x=4\) and \(y=3\) into the algebraic expression which results in \(3(4) - 2(3)\).

Step 3: Compute the result

Now, compute the result of expression. We obtain \(12 - 6\) after the multiplication of variables with their coefficients.

Step 4: Final Computation

After completing the subtractions, we get the final result of \(6\).

Key Concepts

Variable Substitution

Variable substitution is a vital step when working with algebraic expressions. It is the process of replacing variables in an expression with given numerical values. In our example, the expression given is \(3x - 2y\), and we have the values \(x=4\) and \(y=3\).

• First, identify the variables, which are typically represented by letters such as \(x\) and \(y\).
• Next, replace these variables with the specified numbers. So, wherever you see \(x\), you substitute it with \(4\), and for \(y\), you put \(3\).
• This gives you \(3(4) - 2(3)\). The expression now looks like a straightforward arithmetic operation.

Substitution is crucial as it transforms the problem from symbolic form into a numeric one, making it manageable to solve.

Algebra Evaluation

After substituting the variables, the next step is algebra evaluation. This involves simplifying and calculating the numeric value of an expression using order of operations. Let's walk through the expression \(3(4) - 2(3)\).

• Multiply the coefficients by their respective numbers: \(3\times4 = 12\) and \(2\times3 = 6\).
• Write down the simplified expression: \(12 - 6\).
• Finally, subtract the numbers to get the result.

Evaluation consists of breaking down the expression into simpler steps and solving them one by one, focusing on operations in their correct order.

Mathematical Operations

Mathematical operations play a crucial role during the evaluation of algebraic expressions. In the expression \(3(4) - 2(3)\), we apply two main operations: multiplication and subtraction.

• Multiplication: This involves multiplying variables after they've been substituted with their respective numbers. It's essential for modifying the values of \(x\) and \(y\) appropriately based on their coefficients.
• Subtraction: Once the multiplication is done, subtraction helps in finding the difference between the two results. From our example, \(12 - 6\) gives the final answer.

Understanding these operations and the correct sequence to perform them ensures you solve the expression accurately. The order of operations is vital to get the right answer.