Evaluating The Algebraic Expression 36x - 8y^2 Given X = 3 And Y = -6
In this article, we will explore how to evaluate the algebraic expression $36x - 8y^2$ given the values of the variables $x$ and $y$. Specifically, we will substitute $x = 3$ and $y = -6$ into the expression and simplify it to find its numerical value. This type of problem is fundamental in algebra and helps to solidify the understanding of variable substitution and order of operations. By working through this example, we can gain a clearer understanding of how to manipulate algebraic expressions and arrive at accurate solutions.
Understanding the Problem
The core of this problem lies in the ability to correctly substitute the given values into the expression and then follow the order of operations (PEMDAS/BODMAS) to simplify it. Let's break down the expression $36x - 8y^2$. It involves two terms: $36x$ and $-8y^2$. The first term is a simple multiplication of 36 and $x$, while the second term involves an exponent, multiplication, and a negative sign. When substituting $x = 3$ and $y = -6$, it is crucial to handle the exponent and the negative sign correctly.
The Order of Operations (PEMDAS/BODMAS)
To evaluate any mathematical expression correctly, we need to adhere to the order of operations, often remembered by the acronyms PEMDAS or BODMAS:
- Parentheses / Brackets
- Exponents / Orders
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
This order dictates that we must first address any expressions within parentheses or brackets, followed by exponents, then multiplication and division (working from left to right), and finally addition and subtraction (also from left to right). In our case, the exponent in the term $-8y^2$ needs to be dealt with before performing the multiplication.
Substitution of Variables
The substitution process involves replacing the variables in the expression with their given numerical values. In our problem, we have $x = 3$ and $y = -6$. This means we will replace every instance of $x$ with 3 and every instance of $y$ with -6. It's important to use parentheses when substituting negative values to avoid sign errors. After substitution, the expression will look like this: $36(3) - 8(-6)^2$. This sets the stage for the next step, which is simplifying the expression according to the order of operations.
Step-by-Step Evaluation
Now, let's go through the evaluation step-by-step, ensuring we follow the order of operations meticulously. This careful approach is key to avoiding errors and arriving at the correct answer. Each step will be explained in detail to provide a clear understanding of the process.
Step 1: Substitute the Values
The first step, as discussed earlier, is to substitute the given values of $x$ and $y$ into the expression. We have $x = 3$ and $y = -6$. Substituting these values into $36x - 8y^2$ gives us:
This substitution sets the stage for the next operations. Notice the use of parentheses around the substituted values, especially the negative value for $y$, which is crucial for correct evaluation.
Step 2: Evaluate the Exponent
According to the order of operations, we need to address the exponent before any multiplication or subtraction. In our expression, we have $(-6)^2$, which means -6 multiplied by itself:
Substituting this back into our expression, we get:
The exponent is now evaluated, and we can move on to the next operation in the order, which is multiplication.
Step 3: Perform the Multiplications
Now we have two multiplication operations to perform: $36(3)$ and $8(36)$. Let's calculate them:
Substituting these products back into our expression, we get:
We are now left with a simple subtraction problem.
Step 4: Perform the Subtraction
Finally, we perform the subtraction:
Therefore, the value of the expression $36x - 8y^2$ when $x = 3$ and $y = -6$ is -180.
Detailed Explanation of Each Step
To ensure a comprehensive understanding, let's revisit each step with a more detailed explanation. This will help reinforce the concepts and techniques used in the evaluation process.
Detailed Step 1: Substitution
Substitution is the process of replacing variables with their given values. This is a fundamental skill in algebra and is used extensively in solving equations and evaluating expressions. In our case, we are given $x = 3$ and $y = -6$. The expression is $36x - 8y^2$. We replace $x$ with 3 and $y$ with -6. This gives us:
The parentheses are crucial, especially when substituting negative numbers. They ensure that the negative sign is included in the operation, particularly when dealing with exponents. Without the parentheses, we might misinterpret the expression and calculate it incorrectly.
Detailed Step 2: Evaluating the Exponent
The exponent indicates how many times a number is multiplied by itself. In our case, we have $(-6)^2$, which means -6 multiplied by -6. When a negative number is raised to an even power, the result is positive. Therefore:
It's important to remember that $-6^2$ is different from $(-6)^2$. The former means $-(6 imes 6)$, which equals -36, while the latter means $(-6) imes (-6)$, which equals 36. This distinction is crucial in preventing errors.
Detailed Step 3: Multiplication
Multiplication is a basic arithmetic operation that involves finding the product of two or more numbers. In our expression, we have two multiplications to perform: $36 imes 3$ and $8 imes 36$. Performing these multiplications gives us:
These results are then substituted back into the expression, replacing the multiplication operations with their products.
Detailed Step 4: Subtraction
Subtraction is another basic arithmetic operation that involves finding the difference between two numbers. In our final step, we need to subtract 288 from 108:
When subtracting a larger number from a smaller number, the result is negative. In this case, 288 is larger than 108, so the result is a negative number. The difference between 288 and 108 is 180, so the final result is -180.
Common Mistakes to Avoid
Evaluating algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to watch out for:
- Incorrect Order of Operations: Not following the order of operations (PEMDAS/BODMAS) is a common mistake. Make sure to handle exponents before multiplication and division, and multiplication and division before addition and subtraction.
- Sign Errors: Sign errors are particularly common when dealing with negative numbers. Pay close attention to the signs and use parentheses when substituting negative values to avoid confusion.
- Misinterpreting Exponents: As mentioned earlier, be careful to distinguish between $-a^2$ and $(-a)^2$. The former means the negative of $a^2$, while the latter means $(-a)$ squared.
- Arithmetic Errors: Simple arithmetic errors can derail the entire calculation. Double-check your multiplications, divisions, additions, and subtractions to ensure accuracy.
- Forgetting to Substitute: Sometimes, students may forget to substitute the given values for the variables, leaving the expression unevaluated.
By being aware of these common mistakes, you can take steps to avoid them and increase your chances of arriving at the correct answer.
Conclusion
In conclusion, we have successfully evaluated the expression $36x - 8y^2$ when $x = 3$ and $y = -6$. By carefully following the order of operations and paying close attention to the signs, we arrived at the solution of -180. This exercise demonstrates the importance of understanding and applying the fundamental principles of algebra, such as variable substitution and the order of operations. By mastering these concepts, you can confidently tackle more complex algebraic problems and build a strong foundation in mathematics. Remember to always double-check your work and be mindful of common mistakes to ensure accuracy in your calculations. The key takeaway is that practice and attention to detail are essential for success in algebra. This detailed walkthrough should provide a solid understanding of the process and help you tackle similar problems with confidence. The ability to evaluate expressions like this is a critical skill in many areas of mathematics and related fields. By understanding the steps and common pitfalls, you can improve your accuracy and problem-solving abilities.
