Problem 23 - Beginning and Intermediate Algebra

Question

Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(6.459 x\)

Step-by-Step Solution

Answer

(a) 25.836, (b) 38.754

Step 1 - Substitute for Case (a)

For part (a), substitute the value of \(x\) with 4 in the expression \(6.459x\). This gives: \[6.459 \times 4\]

Step 2 - Calculate Case (a)

Multiply 6.459 by 4: \[6.459 \times 4 = 25.836\]

Step 3 - Substitute for Case (b)

For part (b), substitute the value of \(x\) with 6 in the expression \(6.459x\). This gives: \[6.459 \times 6\]

Step 4 - Calculate Case (b)

Multiply 6.459 by 6: \[6.459 \times 6 = 38.754\]

Key Concepts

Substitution Method

Let's dive into the substitution method, a fundamental tool in algebra. The substitution method involves replacing variables with given values to simplify and solve an expression or equation. This technique can be applied to algebraic expressions, equations, and even systems of equations. In our exercise, we were given the expression \(6.459x\) and instructed to evaluate it for two different values of \(x\): 4 and 6.

By substituting these values into the expression, the variable \(x\) is replaced with the given numbers, turning the algebraic expression into a simple arithmetic problem. For instance, in case (a), substituting \(x = 4\) transforms \(6.459x\) into \(6.459 \times 4\). This process helps to understand how changing the value of a variable influences the value of the entire expression.

Here's how you can apply the substitution method:

• Identify the variable(s) in the expression.
• Replace the variable(s) with the given value(s).
• Simplify the resulting expression using arithmetic operations.

Multiplication

Multiplication is one of the basic arithmetic operations. In algebra, multiplication often involves variables and constants. Understanding how to accurately multiply numbers is crucial to evaluate algebraic expressions.

In our exercise, we encountered two multiplication problems: \(6.459 \times 4\) and \(6.459 \times 6\). Here’s a step-by-step approach:

• Align the numbers carefully if you are doing it by hand to ensure accuracy.
• Perform the multiplication operation from right to left.
• Keep track of decimal points. When you multiply decimals, count the total number of decimal places in both numbers to place the decimal in the final product correctly.

For example, in case (a), multiplying 6.459 by 4:

\[ 6.459 \times 4 = 25.836 \]
Similarly, for case (b), multiplying 6.459 by 6:
\[ 6.459 \times 6 = 38.754 \]
Practice makes perfect, so using these steps will help you become more comfortable with multiplication in algebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are the backbone of algebra and are used to represent real-world situations, model problems, and perform calculations.

In our exercise, the algebraic expression given was \(6.459x\), where \(x\) is a variable that can take different values. Important components of algebraic expressions include:

• Variables: Symbols representing unknown values (e.g., \(x\), \(y\)).
• Constants: Fixed values (e.g., 6.459).
• Operations: Mathematical operations such as addition, subtraction, multiplication, and division (e.g., \(6.459x\) implies multiplication).

Evaluating algebraic expressions involves substituting variables with given numbers and then simplifying the expression using arithmetic operations. This allows you to see how the value of the expression changes with different variable values.

Understanding and practicing the manipulation of algebraic expressions is crucial for solving more complex algebra problems and helps build a strong foundation in mathematics.