Evaluate 3z When Z 9

Evaluating 3z When z = 9: A Deep Dive into Mathematical Operations

This article provides a comprehensive explanation of how to evaluate the algebraic expression 3z when z is equal to 9. While seemingly simple, this problem serves as a fundamental building block for understanding more complex algebraic manipulations and problem-solving techniques. We will explore the concept of algebraic substitution, delve into the order of operations (often remembered by the acronym PEMDAS/BODMAS), and discuss the broader implications of this seemingly straightforward calculation. Understanding this process is crucial for success in algebra and beyond.

Introduction to Algebraic Expressions and Substitution

In mathematics, an algebraic expression is a combination of variables, constants, and mathematical operations (like addition, subtraction, multiplication, and division). A variable, often represented by a letter (like z in our case), represents an unknown value or a value that can change. A constant is a fixed numerical value.

The expression 3z is a simple algebraic expression. It indicates that a variable z is being multiplied by the constant 3. To evaluate this expression, we need to substitute a specific value for the variable z. In this case, we are given that z = 9. Substitution involves replacing the variable with its given numerical value.

Step-by-Step Evaluation of 3z When z = 9

Evaluating 3z when z = 9 is a straightforward process:

1. Substitute: Replace the variable z with the value 9 in the expression 3z. This gives us 3 * 9.

2. Multiply: Perform the multiplication operation: 3 * 9 = 27.

Therefore, the value of 3z when z = 9 is 27.

The Importance of Order of Operations (PEMDAS/BODMAS)

While this particular example is simple, understanding the order of operations is crucial when evaluating more complex algebraic expressions. The order of operations dictates the sequence in which mathematical operations should be performed. This is often remembered using the acronyms PEMDAS or BODMAS:

• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• BODMAS: Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Both acronyms represent the same order of operations; the only difference lies in the terminology used. "Parentheses" and "Brackets" are interchangeable, as are "Exponents" and "Orders."

Let's illustrate the importance of PEMDAS/BODMAS with a slightly more complex example:

Evaluate 2 * z + 5 when z = 9.

Following PEMDAS/BODMAS:

1. Substitution: Replace z with 9: 2 * 9 + 5

2. Multiplication: Perform the multiplication first: 18 + 5

3. Addition: Perform the addition: 23

Therefore, the value of 2 * z + 5 when z = 9 is 23. If we had performed the addition before the multiplication, we would have obtained an incorrect result.

Expanding the Concept: Working with More Complex Expressions

The fundamental principle of substitution remains the same even when dealing with more complex algebraic expressions. Consider the expression:

2z² + 4z - 7 where z = 9

The steps involved are:

1. Substitution: Replace each instance of z with 9: 2(9)² + 4(9) - 7

2. Exponents: Evaluate the exponent first: 2(81) + 4(9) - 7

3. Multiplication: Perform the multiplications: 162 + 36 - 7

4. Addition and Subtraction: Perform addition and subtraction from left to right: 198 - 7 = 191

Therefore, the value of 2z² + 4z - 7 when z = 9 is 191.

Real-World Applications: Where This Matters

The ability to evaluate algebraic expressions is fundamental to numerous real-world applications across various fields:

• Physics: Calculating velocity, acceleration, or force using equations that involve variables.

• Engineering: Designing structures, circuits, or systems where variables represent dimensions, currents, or pressures.

• Finance: Calculating interest, compound growth, or investment returns based on formulas involving variables.

• Computer Science: Programming algorithms and writing code that involves manipulating variables and performing calculations.

• Economics: Modeling economic systems and predicting trends based on equations that involve variables representing economic indicators.

Beyond the Basics: Developing Algebraic Fluency

Mastering the evaluation of algebraic expressions is a cornerstone of algebraic fluency. Fluency goes beyond simply getting the right answer; it encompasses understanding the underlying principles, recognizing patterns, and applying those principles to increasingly complex problems. Regular practice and a focus on understanding why the steps are performed in a certain order are key to building this fluency.

Frequently Asked Questions (FAQ)

Q: What happens if I substitute the value of z incorrectly?

A: Substituting the value of z incorrectly will lead to an incorrect final answer. Accuracy in substitution is paramount. Always double-check your work to ensure you've replaced z with the correct value (9, in this case).

Q: Can I solve this problem using a calculator?

A: Yes, a calculator can significantly speed up the process, particularly with more complex expressions. However, it's crucial to understand the underlying mathematical principles and the order of operations before relying solely on a calculator. Calculators can help with the computation but shouldn't replace understanding.

Q: What if the expression involved other operations besides multiplication?

A: The principles remain the same. You still substitute the value of z and then follow the order of operations (PEMDAS/BODMAS) to correctly evaluate the expression, regardless of the operations involved (addition, subtraction, division, exponents, etc.).

Q: Are there other ways to represent this problem?

A: Yes, the problem could be presented in word problems. For example: "John earns $3 for every item he sells. If he sells 9 items, how much money does he earn?" This word problem translates directly into the algebraic expression 3z where z represents the number of items sold.

Conclusion: Mastering the Fundamentals

Evaluating 3z when z = 9 might seem trivial, but it encapsulates fundamental concepts in algebra – substitution, order of operations, and the importance of accuracy. Mastering these seemingly simple concepts is crucial for building a solid foundation in algebra and tackling more advanced mathematical problems. Remember that consistent practice and a focus on understanding the underlying principles are key to achieving algebraic fluency and successfully applying these skills in diverse contexts. By understanding the "why" behind the calculations, you'll not only solve problems correctly but also gain a deeper appreciation for the power and elegance of mathematics.