Problem 74 - Introductory Algebra for College Students

Question

In Exercises \(73-80,\) evaluate each algebraic expression for the given value of the variable. $$x^{2}-2 x ; x=6$$

Step-by-Step Solution

Answer

The result of evaluating the algebraic expression \(x^{2}-2x\) for \(x=6\) is 24.

Step 1: Substitution

Substitute the given value of x into the algebraic expression. The expression \(x^{2}-2x\) becomes \(6^{2}-2*6\).

Step 2: Squaring

Calculate the square of 6. \(6^{2} = 36.\) This leaves us with the expression \(36-2*6\).

Step 3: Multiplication

Perform the multiplication operation next, according to the BIDMAS/BODMAS rules. Hence, \(2*6 = 12\). This leaves us with the expression \(36-12\).

Step 4: Subtraction

Finally, subtract 12 from 36. This results in \(36-12=24\).

Key Concepts

Variable Substitution

Variable substitution is the process of replacing a variable in an algebraic expression with a given numerical value. Essentially, it allows us to evaluate expressions by converting them into simpler arithmetic operations. In the given exercise, we substitute the variable \(x = 6\) into the algebraic expression \(x^2 - 2x\).

• First, identify the variable present in the expression, which in this case is \(x\).
• Next, substitute \(x\) with the given number, \(6\), resulting in a new expression \(6^2 - 2 \times 6\).

The process of variable substitution transforms the algebraic problem into something we can compute using basic arithmetic methods.
This step sets the stage for further simplification using the order of operations. By substituting the variable with its assigned value, we make the expression ready for calculation.

Order of Operations

Order of operations is critical to correctly solving any algebraic expression following the substitution of variables. It refers to the sequence in which mathematical operations should be done to ensure consistent and correct results. Known by acronyms such as BIDMAS or BODMAS, the order is:

• Brackets
• Indices (such as squares and roots)
• Division and Multiplication (performed from left to right)
• Addition and Subtraction (also performed from left to right)

In the example given, after the substitution of \(x\), the expression becomes \(6^2 - 2 \times 6\). According to BIDMAS, the index comes first, so we compute the exponent:\[6^2 = 36\]
Next, we perform the multiplication \(2 \times 6 = 12\), as multiplication precedes subtraction.
Finally, after completing these operations, we are left with a simple subtraction: \[36 - 12\]. The order of operations ensures that each step is executed in the right sequence for accurate results.

Evaluating Expressions

Once variable substitution and order of operations have been correctly applied, evaluating expressions becomes straightforward. Evaluating an expression involves simplifying it to its final numerical result. Let's summarize the steps used in evaluating the given expression:

• Begin by substituting the variable \(x\) with the value \(6\), giving you \(6^2 - 2 \times 6\).
• Use the correct order of operations: calculate the exponent, perform multiplication, and then subtraction.
• From \(6^2 = 36\) and \(2 \times 6 = 12\), we reach \(36 - 12\).
• Finally, perform the subtraction to evaluate the expression completely: \[36 - 12 = 24\].

Completing these steps leads to the final result of the expression "24", highlighting the importance of both variable substitution and order of operations in solving algebraic expressions efficiently.