Calculate Y Step-by-Step Using Order Of Operations
In this article, we will walk through the process of calculating the value of y in the equation y = 2(a - b)² - c, given the values a = -11, b = -5, and c = 19. We will meticulously follow the order of operations (PEMDAS/BODMAS) to ensure accurate simplification and rounding to two decimal places where necessary. Understanding the order of operations is crucial in mathematics, and this example provides a practical application of these principles.
Understanding the Order of Operations
The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed. This ensures that mathematical expressions are evaluated consistently, leading to a single correct answer. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) are commonly used to remember the correct order.
- Parentheses/Brackets: Operations inside parentheses or brackets are performed first.
- Exponents/Orders: Next, we evaluate exponents and roots.
- Multiplication and Division: Multiplication and division are performed from left to right.
- Addition and Subtraction: Finally, addition and subtraction are performed from left to right.
By adhering to this order, we can break down complex expressions into manageable steps, ensuring accuracy and clarity in our calculations. Now, let's apply these rules to our equation.
Step-by-Step Calculation
Let's break down the calculation of y = 2(a - b)² - c, given a = -11, b = -5, and c = 19, into manageable steps:
1. Substitute the Given Values
Our first step involves substituting the given values of a, b, and c into the equation:
y = 2(a - b)² - c
Substituting a = -11, b = -5, and c = 19, we get:
y = 2((-11) - (-5))² - 19
This substitution sets the stage for the subsequent operations, ensuring we work with the correct numerical values.
2. Simplify Inside the Parentheses
Next, we focus on the operation inside the parentheses. According to the order of operations, parentheses are prioritized.
y = 2((-11) - (-5))² - 19
To simplify (-11) - (-5), we recall that subtracting a negative number is equivalent to adding its positive counterpart:
(-11) - (-5) = -11 + 5
Now, we add -11 and 5:
-11 + 5 = -6
So, the expression inside the parentheses simplifies to -6:
y = 2(-6)² - 19
This step reduces the complexity of the equation, making it easier to proceed with the remaining operations.
3. Evaluate the Exponent
Now, we address the exponent. The term (-6)² means -6 raised to the power of 2, which is -6 multiplied by itself.
y = 2(-6)² - 19
Calculating (-6)²:
(-6)² = (-6) * (-6) = 36
So, our equation becomes:
y = 2(36) - 19
Evaluating the exponent is a crucial step in simplifying the expression, as it sets the stage for the multiplication that follows.
4. Perform Multiplication
With the exponent evaluated, we now perform the multiplication. We multiply 2 by 36:
y = 2(36) - 19
Multiplying 2 and 36:
2 * 36 = 72
Our equation now looks like this:
y = 72 - 19
Performing the multiplication simplifies the equation further, leaving us with a simple subtraction to complete.
5. Perform Subtraction
Finally, we perform the subtraction to find the value of y.
y = 72 - 19
Subtracting 19 from 72:
72 - 19 = 53
Therefore, the value of y is 53.
This final step completes the calculation, giving us the solution to the equation.
Final Result
After meticulously following the order of operations and performing each step, we have arrived at the final result.
Given a = -11, b = -5, and c = 19, we substituted these values into the equation y = 2(a - b)² - c and simplified it step by step.
The final value of y is:
y = 53
This result demonstrates the importance of adhering to the order of operations to ensure accuracy in mathematical calculations.
Conclusion
In summary, we have successfully calculated the value of y by substituting the given values of a, b, and c into the equation y = 2(a - b)² - c. We meticulously followed the order of operations (PEMDAS/BODMAS), which includes prioritizing parentheses, exponents, multiplication, and subtraction in that specific sequence.
This exercise highlights the significance of understanding and applying the order of operations to solve mathematical expressions accurately. By breaking down the problem into smaller, manageable steps, we were able to simplify the equation and arrive at the correct answer. The final result, y = 53, demonstrates the practical application of these mathematical principles.
By mastering the order of operations, one can confidently tackle various mathematical problems and ensure consistent and accurate results. This fundamental skill is essential for success in mathematics and related fields. Remember to always prioritize the order of operations to maintain precision and clarity in your calculations.
