Evaluating (6x - 3y)^2 For X=5 And Y=4 A Step-by-Step Guide
This article provides a detailed, step-by-step explanation of how to evaluate the algebraic expression (6x - 3y)^2 when given the values of x and y. Specifically, we'll focus on the case where x equals 5 and y equals 4. The value of z (z=4) is provided in the prompt, but it's important to note that z does not appear in the expression we are evaluating, so we won't need to use it in our calculation. This exploration will help solidify your understanding of algebraic substitution and the order of operations.
Understanding Algebraic Expressions
Before we jump into the calculation, let's briefly revisit what an algebraic expression is. An algebraic expression is a combination of variables (like x and y), constants (numbers), and mathematical operations (like addition, subtraction, multiplication, division, and exponentiation). Evaluating an algebraic expression means finding its numerical value when we substitute specific numbers for the variables. The expression (6x - 3y)^2 fits this definition perfectly. It involves the variables x and y, the constants 6 and 3, subtraction, multiplication (implied between the constants and the variables), and exponentiation (the power of 2). To accurately evaluate this expression, we must adhere to the order of operations, commonly remembered by the acronym PEMDAS or BODMAS.
The Order of Operations: PEMDAS/BODMAS
The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed. PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) are two acronyms that help us remember this order. They both represent the same hierarchy:
- Parentheses / Brackets: Operations within parentheses or brackets are performed first.
- Exponents / Orders: Exponents (powers) and roots are evaluated next.
- Multiplication and Division: These operations are performed from left to right.
- Addition and Subtraction: These operations are performed from left to right.
Following this order is crucial to arrive at the correct answer when evaluating any mathematical expression. We will apply PEMDAS in our calculation of (6x - 3y)^2.
Step-by-Step Evaluation
Now, let's evaluate the expression (6x - 3y)^2 for x = 5 and y = 4. We will meticulously follow the order of operations to ensure accuracy.
Step 1: Substitution
The first step is to substitute the given values of x and y into the expression. Replace x with 5 and y with 4:
(6 * 5 - 3 * 4)^2
This substitution transforms the algebraic expression into a purely numerical one.
Step 2: Parentheses
According to PEMDAS, we need to evaluate the expression within the parentheses first. Inside the parentheses, we have two multiplication operations and one subtraction. We perform the multiplication operations before the subtraction.
(6 * 5 - 3 * 4)^2 becomes (30 - 12)^2
Next, we perform the subtraction within the parentheses:
(30 - 12)^2 becomes (18)^2
We have now simplified the expression inside the parentheses to a single number.
Step 3: Exponent
The next operation in PEMDAS is the exponent. We have (18)^2, which means 18 raised to the power of 2, or 18 multiplied by itself:
(18)^2 becomes 18 * 18
Calculating 18 multiplied by 18, we get:
18 * 18 = 324
Step 4: Final Result
After performing the exponentiation, we have arrived at the final result. There are no further operations to perform.
Therefore, the value of the expression (6x - 3y)^2 when x = 5 and y = 4 is 324.
Conclusion
In this article, we successfully evaluated the algebraic expression (6x - 3y)^2 for the given values of x = 5 and y = 4. By meticulously following the order of operations (PEMDAS), we substituted the values, simplified the expression within the parentheses, and then calculated the exponent. This process highlights the importance of understanding and applying the order of operations to ensure accurate results in algebraic evaluations. The final answer, 324, represents the numerical value of the expression under the specified conditions. Remember that the variable z and its value (z = 4) were not relevant to this particular calculation, as z did not appear in the original expression. This exercise provides a solid foundation for tackling more complex algebraic problems. Practicing these steps will improve your proficiency in algebra and your ability to solve mathematical problems accurately and efficiently.
In the realm of algebra, evaluating expressions is a fundamental skill. This article provides a comprehensive, step-by-step guide on how to evaluate the expression (6x - 3y)^2 when given the values x = 5 and y = 4. While the prompt also includes z = 4, it's important to recognize that z is not present in the expression, making it irrelevant to our calculation. This guide aims to solidify your understanding of algebraic substitution, the order of operations, and the importance of accurate calculation in mathematics. We will break down each step, providing clear explanations and insights to help you master this essential algebraic technique.
Grasping the Concept of Algebraic Evaluation
Algebraic evaluation is the process of determining the numerical value of an expression by substituting specific values for the variables it contains. An algebraic expression is a combination of variables (represented by letters like x, y, and z), constants (numbers), and mathematical operations (+, -, *, /, ^). The expression (6x - 3y)^2 is a prime example. It involves the variables x and y, the constants 6 and 3, multiplication (implied between the constants and variables), subtraction, and exponentiation (the power of 2). To correctly evaluate such expressions, we must adhere to the established order of operations, often remembered by the acronyms PEMDAS or BODMAS. This order ensures that we perform the operations in the correct sequence, leading to the accurate numerical result.
The Indispensable Order of Operations: PEMDAS/BODMAS Explained
The order of operations, represented by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is the cornerstone of mathematical calculations. Both acronyms represent the same hierarchical structure, ensuring consistent and accurate results across different mathematical problems. Let's delve deeper into each component:
- Parentheses / Brackets: Operations enclosed within parentheses or brackets take precedence. These grouping symbols indicate that the enclosed expressions should be simplified first, isolating them from the rest of the equation.
- Exponents / Orders: Exponents, which represent repeated multiplication (e.g., x^2 means x * x), and roots (the inverse operation of exponents) are evaluated next. They signify the power or degree to which a number is raised.
- Multiplication and Division: These operations hold equal priority and are performed from left to right. This means that in an expression containing both multiplication and division, the operation encountered first (reading from left to right) is executed first.
- Addition and Subtraction: Similar to multiplication and division, addition and subtraction share the same level of precedence and are performed from left to right. The order in which they appear in the expression dictates their sequence of execution.
Adhering strictly to this order is paramount in obtaining the correct answer. Deviating from PEMDAS/BODMAS can lead to erroneous results, especially in complex expressions. We will meticulously apply this order throughout our evaluation of (6x - 3y)^2, ensuring accuracy at every step.
A Detailed Step-by-Step Evaluation Process
Let's embark on the evaluation of the expression (6x - 3y)^2 for x = 5 and y = 4. Our approach will be methodical, following the order of operations to guarantee a precise outcome.
Step 1: The Art of Substitution
The initial step involves replacing the variables in the expression with their assigned numerical values. We substitute x with 5 and y with 4, transforming the algebraic expression into a purely numerical one:
(6 * 5 - 3 * 4)^2
This substitution is the bridge between the abstract algebraic form and the concrete numerical calculation.
Step 2: Navigating the Parentheses
Following PEMDAS, our focus now shifts to the expression within the parentheses. Inside, we encounter two multiplication operations and one subtraction. Multiplication takes precedence over subtraction, so we execute the multiplications first:
(6 * 5 - 3 * 4)^2 transforms into (30 - 12)^2
Next, we perform the subtraction within the parentheses:
(30 - 12)^2 simplifies to (18)^2
We have successfully reduced the expression within the parentheses to a single numerical value.
Step 3: Unleashing the Exponent
The next operation in the hierarchy is the exponent. We have (18)^2, which signifies 18 raised to the power of 2, or 18 multiplied by itself:
(18)^2 becomes 18 * 18
Performing the multiplication, we obtain:
18 * 18 = 324
The exponentiation operation yields our intermediate result.
Step 4: The Grand Finale – The Final Result
Having executed all operations according to PEMDAS, we arrive at our final result. No further calculations are required.
Therefore, the value of the expression (6x - 3y)^2 when x = 5 and y = 4 is 324.
Conclusion: Mastering Algebraic Evaluation
In this comprehensive guide, we meticulously evaluated the algebraic expression (6x - 3y)^2 for x = 5 and y = 4. By strictly adhering to the order of operations (PEMDAS), we navigated the substitution, parentheses simplification, and exponentiation, ultimately arriving at the solution: 324. It's worth reiterating that the presence of z = 4 in the original prompt was a distraction, as z did not appear in the expression and thus had no impact on the calculation. This exercise underscores the critical importance of understanding and applying the order of operations to ensure accuracy in algebraic evaluations. The methodical approach demonstrated here serves as a solid foundation for tackling more complex algebraic challenges. Consistent practice of these steps will enhance your algebraic proficiency and your ability to solve mathematical problems with confidence and precision.
This meticulous process highlights the core principles of algebraic evaluation and provides a framework for approaching similar problems with confidence.
