Solving the Expression (2xy - frac{5}{2}) with Given Values
In this article, we will evaluate the algebraic expression (2xy - frac{5}{2}) given the values (x 3) and (y 4). We will solve the problem using both the BODMAS and PEMDAS rules, which are used to determine the order of operations in mathematics. By following these steps, readers will gain a deeper understanding of how to manipulate and evaluate algebraic expressions.
Order of Operations: BODMAS vs PEMDAS
The order of operations is a fundamental concept in mathematics that ensures we perform calculations correctly. Two common mnemonics used to remember the order of operations are BODMAS and PEMDAS. Both acronyms stand for the same set of operations, but they are just represented in different orders:
BODMAS: Brackets, Orders (powers and roots, etc.), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right) PEMDAS: Parentheses, Exponents (powers and roots, etc.), Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)For the purpose of clarity, we will use PEMDAS, as it is more widely used in the United States. However, the results using BODMAS would be identical in this case.
Solving the Given Expression
Given the expression (2xy - frac{5}{2}) and the values (x 3) and (y 4), we will substitute these values into the expression and then evaluate it step-by-step.
Step-by-Step Solution Using PEMDAS
1. Multiplication: First, we perform the multiplication (2 cdot 3 cdot 4).
(2 cdot 3 cdot 4 24)
2. Subtraction and Division: Next, we perform the division (frac{5}{2}).
(frac{5}{2} 2.5)
3. Subtraction: Finally, we subtract 2.5 from 24.
24 - 2.5 21.5
Thus, the value of the expression (2xy - frac{5}{2}) when (x 3) and (y 4) is 21.5.
Clarification of Miscalculation
The original solution given in the problem statement contains a mistake in the subtraction step. Here's the correct step-by-step evaluation:
Correct Solution
1. Substituting the values (x 3) and (y 4) into the expression (2xy - frac{5}{2}), we get:
(2 cdot 3 cdot 4 - frac{5}{2})
2. Performing the multiplication:
(24 - frac{5}{2})
3. Converting (frac{5}{2}) to a decimal for easier subtraction:
(24 - 2.5)
4. Subtracting 2.5 from 24:
24 - 2.5 21.5
Therefore, the correct value of the expression is 21.5.
Conclusion
Through the use of the correct order of operations, we have evaluated the expression (2xy - frac{5}{2}) given the values (x 3) and (y 4) and found the result to be 21.5. This exercise demonstrates the importance of following mathematical principles and highlights the potential pitfalls when making arithmetic mistakes.
Frequently Asked Questions (FAQs)
What is the importance of the order of operations?
The order of operations is crucial in mathematics because it ensures that everyone arrives at the same answer for a given expression. Without a standard set of rules, different people might solve the same problem differently, leading to inconsistent results.
Why should I learn both BODMAS and PEMDAS?
While both acronyms represent the same set of operations, they can help reinforce your understanding of the order of operations. Learning both can make you more flexible and adaptable when working with different types of problems or in different countries where one mnemonic may be more commonly used than the other.
Can I use any other mnemonic for the order of operations?
Yes, other mnemonics for the order of operations include CADMUS (Careful Addition Before Multiplication and Subtraction), GEMDAS (Grouping, Exponents, Multiplication, Division, Addition, Subtraction), and even BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction). However, BODMAS and PEMDAS are the most widely recognized.
What are some real-world applications of algebraic expressions?
Algebraic expressions have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, they can be used to model complex systems, optimize processes, and make predictions. Understanding how to evaluate algebraic expressions accurately is a fundamental skill in these and many other domains.
References
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