Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression when \(x=3, y=-2\), and \(z=4$$z^{2}\) ? \(6 y-x\)

Step-by-Step Solution

Answer

The value of the given expression for \(x=3\), \(y=-2\), and \(z=4\) is 25.

Step 1: Substitute the given values into the expression

Replace \(x\), \(y\), and \(z\) in the equation \(z^{2} - 6y -x\) by their respective values given: \(x=3\), \(y=-2\), \(z=4\). This results in \(4^{2} - 6*(-2) -3\).

Step 2: Apply the order of operations

First perform the operation of exponentiation, which results in \(4^{2} = 16\). Next, do the multiplication resulting in \(-6*(-2) = 12\). So, the equation now looks like: \(16 + 12 -3\).

Step 3: Perform the addition and subtraction

By adding and subtracting the numbers, we find that the expression amounts to \(16 + 12 - 3 = 25\).

Key Concepts

Substitution in Algebra

Substitution in algebra is a foundational concept, making it easier to solve expressions and equations by replacing variables with their given values. This technique simplifies complex algebraic expressions, allowing us to work with numbers instead of letters. In the original exercise, the problem asks us to substitute the values of \(x\), \(y\), and \(z\) into the expression \(z^{2} - 6y -x\). This means:

• Replace \(z\) with 4, so \(z^2\) becomes \(4^2\).
• Replace \(y\) with -2, so \(6y\) becomes \(6*(-2)\).
• Replace \(x\) with 3, so \( - x\) becomes \(- 3\).

By substituting these values, we transform our abstraction into a numeric form: \(4^{2} - 6*(-2) -3\). Substitution thus simplifies the task and sets the stage for further calculations.

Order of Operations

The order of operations is crucial for solving algebraic expressions correctly. It dictates the sequence in which mathematical operations should be performed to ensure consistent and accurate results. The correct sequence follows the PEMDAS rule:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

In our simplified expression \(4^{2} - 6(-2) -3\), we first deal with the exponent. Calculate \(4^2\) which results in 16. Next, handle the multiplication: \(-6*(-2)\), providing us with 12. Arrange the terms as \(16 + 12 - 3\) to move onto addition and subtraction, completing the operations in the correct order to avoid mistakes.

Evaluating Expressions

Evaluating expressions involves plugging in the values and performing arithmetic operations to reach a final numerical result. In essence, it tests our understanding of substitution and the order of operations. For the given expression and its values, we proceed with:

• Substitute the variables with their values: makes the expression numerical.
• Follow order of operations with precision: maintains accuracy and minimizes errors.
• Calculate each step until simplification: carry out arithmetic operations carefully.

For the expression \(16 + 12 - 3\), simply add 16 and 12 to get 28, then subtract 3 to finalise it as 25. Evaluating an expression effectively converts it from variables and operators to a simple number, offering clarity and understanding.