Evaluating The Expression (4x-12) + (1/2xy - 10) For X=4 And Y=6
In the realm of mathematics, evaluating expressions is a fundamental skill. It allows us to understand the behavior of equations and functions for specific input values. This article delves into the process of finding the value of the expression (4x - 12) + (1/2xy - 10) when x equals 4 and y equals 6. By meticulously substituting these values and performing the necessary arithmetic operations, we can unravel the solution. This exploration not only reinforces our understanding of algebraic manipulation but also highlights the practical application of mathematical principles in problem-solving.
Understanding the Expression
The expression we are tasked with evaluating is (4x - 12) + (1/2xy - 10). This expression combines algebraic terms involving the variables x and y, along with constant terms. To accurately find its value for x = 4 and y = 6, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform the operations in the correct sequence, leading to the correct answer. Understanding the structure of the expression and the order of operations is crucial for successful evaluation.
Breaking Down the Components
To effectively tackle the expression, let's break it down into its constituent parts. The expression consists of two main parts enclosed in parentheses: (4x - 12) and (1/2xy - 10). The first part, (4x - 12), involves multiplying the variable x by 4 and then subtracting 12. The second part, (1/2xy - 10), involves multiplying x and y, halving the result, and then subtracting 10. The entire expression is the sum of these two parts. By identifying these individual components, we can systematically substitute the given values and perform the calculations.
The Role of Variables
Variables, such as x and y in our expression, are placeholders for numerical values. They allow us to represent quantities that can change or vary. In this case, we are given specific values for x and y, namely x = 4 and y = 6. Substituting these values into the expression replaces the variables with their corresponding numbers, transforming the algebraic expression into a numerical one. This substitution is the first step in evaluating the expression, as it sets the stage for performing the arithmetic operations. The ability to work with variables is fundamental to algebra and allows us to generalize mathematical relationships.
Substitution: Replacing Variables with Values
The cornerstone of evaluating an algebraic expression for specific values lies in the process of substitution. This involves replacing each variable in the expression with its assigned numerical value. In our case, we are given x = 4 and y = 6. Therefore, we will substitute 4 for every instance of x and 6 for every instance of y within the expression (4x - 12) + (1/2xy - 10). This meticulous replacement is crucial, as even a minor error in substitution can lead to an incorrect final result. Once the variables are replaced, the expression transforms into a purely numerical one, ready for simplification using the order of operations.
Performing the Substitution
Let's perform the substitution step by step. We begin with the expression (4x - 12) + (1/2xy - 10). Replacing x with 4 and y with 6, we get: (4(4) - 12) + (1/2(4)(6) - 10). It is essential to maintain the parentheses and the order of terms as we substitute. This ensures that the structure of the expression remains intact, and we can proceed with the calculations accurately. The expression now contains only numbers and arithmetic operations, setting the stage for the next phase of evaluation.
Ensuring Accuracy in Substitution
The accuracy of the substitution step is paramount to obtaining the correct final value. A small oversight, such as substituting a value for the wrong variable or misplacing a number, can lead to a significantly different result. To minimize errors, it's helpful to double-check each substitution as you make it. You might even want to write out the original expression and the substituted expression side by side, carefully aligning the terms to ensure a one-to-one correspondence. This attention to detail is a hallmark of careful mathematical work and contributes to reliable problem-solving.
Applying the Order of Operations (PEMDAS)
With the variables successfully substituted, the next critical step is to apply the order of operations, often remembered by the acronym PEMDAS. PEMDAS dictates the sequence in which mathematical operations must be performed: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Adhering to this order is crucial for achieving the correct result. Failing to follow PEMDAS can lead to miscalculations and an incorrect final answer. In our expression, we will first address the operations within the parentheses, then perform any multiplication and division, and finally handle addition and subtraction.
Step-by-Step Calculation within Parentheses
Following PEMDAS, we first focus on the operations within the parentheses in our expression: (4(4) - 12) + (1/2(4)(6) - 10). Let's tackle the first set of parentheses, (4(4) - 12). Within these parentheses, we first perform the multiplication: 4 multiplied by 4 equals 16. This gives us (16 - 12). Next, we perform the subtraction: 16 minus 12 equals 4. Thus, the first set of parentheses simplifies to 4.
Now, let's move on to the second set of parentheses, (1/2(4)(6) - 10). Here, we first perform the multiplication from left to right. 1/2 multiplied by 4 equals 2. Then, 2 multiplied by 6 equals 12. This gives us (12 - 10). Finally, we perform the subtraction: 12 minus 10 equals 2. So, the second set of parentheses simplifies to 2.
By meticulously following the order of operations within each set of parentheses, we have simplified the expression to 4 + 2. This sets the stage for the final addition step.
Final Addition: Reaching the Solution
After simplifying the expressions within the parentheses, we are left with the simple addition problem: 4 + 2. Adding these two numbers together, we arrive at the final answer: 6. Therefore, the value of the expression (4x - 12) + (1/2xy - 10) when x = 4 and y = 6 is 6. This result demonstrates the power of algebraic manipulation and the importance of following the order of operations to arrive at the correct solution. By breaking down the problem into manageable steps and applying the rules of mathematics, we have successfully evaluated the expression.
Conclusion: The Value of Order and Precision
In conclusion, we have successfully determined the value of the expression (4x - 12) + (1/2xy - 10) for x = 4 and y = 6. By meticulously substituting the given values and adhering to the order of operations (PEMDAS), we arrived at the solution of 6. This exercise highlights the importance of precision and methodical execution in mathematical problem-solving. Each step, from substitution to simplification, plays a crucial role in achieving the correct result. This process not only reinforces our understanding of algebraic concepts but also showcases the practical application of mathematical principles in evaluating expressions.
The ability to evaluate expressions is a fundamental skill in mathematics and has wide-ranging applications in various fields, from science and engineering to finance and computer science. By mastering this skill, we equip ourselves with a powerful tool for solving complex problems and making informed decisions based on numerical data. This example serves as a reminder of the importance of a solid foundation in basic mathematical principles and the value of applying them with care and accuracy.
