Solving Mathematical Equations Step By Step 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19

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In the realm of mathematics, equations often present themselves as intriguing puzzles, challenging us to decipher the hidden values and relationships within. One such puzzle is the equation 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19, which appears deceptively complex at first glance. However, with a systematic approach and a clear understanding of the order of operations, we can unravel this equation and determine the missing value.

Understanding the Order of Operations

Before we delve into the intricacies of the equation, it's crucial to establish a firm grasp of the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This set of rules dictates the sequence in which mathematical operations must be performed to arrive at the correct solution.

In our equation, we encounter several operations: division, addition, subtraction, and parentheses. According to PEMDAS, we must first address the operations within the parentheses, followed by division, and then finally addition and subtraction.

Step-by-Step Solution

Let's break down the equation step-by-step, applying the order of operations to guide our way:

  1. Simplify the Innermost Parentheses:

    We begin by simplifying the innermost parentheses, (9 + 9), which evaluates to 18. Our equation now transforms to:

    28 - [65 ÷ 5 + 18 - (6 - _)] = 19

  2. Address the Remaining Parentheses:

    Next, we tackle the remaining parentheses, (6 - _). However, we cannot directly evaluate this expression because it contains the missing value, represented by the underscore. Therefore, we'll leave it as is for now and proceed to the next operation.

  3. Perform the Division:

    According to PEMDAS, division takes precedence over addition and subtraction. So, we perform the division operation, 65 ÷ 5, which yields 13. The equation now becomes:

    28 - [13 + 18 - (6 - _)] = 19

  4. Simplify the Brackets:

    Now, we focus on simplifying the expression within the brackets. We perform the addition and subtraction operations from left to right:

    • 13 + 18 = 31
    • 31 - (6 - _)

    Our equation now looks like this:

    28 - [31 - (6 - _)] = 19

  5. Isolate the Missing Value:

    To isolate the missing value, we need to eliminate the brackets and the subtraction operation outside them. We can do this by subtracting 28 from both sides of the equation:

    28 - [31 - (6 - _)] - 28 = 19 - 28

    This simplifies to:

    -[31 - (6 - _)] = -9

  6. Remove the Negative Sign:

    To get rid of the negative sign on the left side, we multiply both sides of the equation by -1:

    -[31 - (6 - _)] * -1 = -9 * -1

    This gives us:

    31 - (6 - _) = 9

  7. Isolate the Parentheses:

    Next, we subtract 31 from both sides of the equation to isolate the parentheses:

    31 - (6 - _) - 31 = 9 - 31

    This simplifies to:

    -(6 - _) = -22

  8. Remove the Negative Sign Again:

    We multiply both sides of the equation by -1 to eliminate the negative sign on the left side:

    -(6 - _) * -1 = -22 * -1

    This results in:

    6 - _ = 22

  9. Solve for the Missing Value:

    Finally, we subtract 6 from both sides of the equation to solve for the missing value:

    6 - _ - 6 = 22 - 6

    This gives us:

    - _ = 16

    Multiplying both sides by -1, we find the missing value:

    _ = -16

Verification

To ensure the accuracy of our solution, we can substitute the value of -16 back into the original equation and verify that it holds true:

28 - [65 ÷ 5 + (9 + 9) - (6 - (-16))] = 19

Following the order of operations:

  • 28 - [65 ÷ 5 + (9 + 9) - (6 + 16)] = 19
  • 28 - [65 ÷ 5 + 18 - 22] = 19
  • 28 - [13 + 18 - 22] = 19
  • 28 - [31 - 22] = 19
  • 28 - 9 = 19
  • 19 = 19

As we can see, the equation holds true when we substitute -16 for the missing value. Therefore, our solution is correct.

Conclusion

The equation 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19 initially appeared daunting, but by systematically applying the order of operations and breaking down the problem into smaller, manageable steps, we were able to successfully solve for the missing value. This exercise highlights the importance of a structured approach and a solid understanding of mathematical principles in tackling complex equations. The missing value in the equation is -16.

Unlocking the secrets of complex mathematical equations often hinges on a solid grasp of the order of operations. Equations like 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19 can seem intimidating at first glance, but with a methodical approach and a clear understanding of the rules, we can systematically solve for the unknown. In this article, we will delve deep into this equation, dissecting each step and highlighting the critical role of the order of operations in arriving at the correct solution.

The Cornerstone: Order of Operations (PEMDAS/BODMAS)

Before we embark on our mathematical journey, let's solidify our understanding of the guiding principle: the order of operations. This principle, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), provides a roadmap for tackling mathematical expressions.

The order of operations dictates the sequence in which mathematical operations should be performed: Parentheses/Brackets first, followed by Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Adhering to this order ensures consistency and accuracy in our calculations.

In our equation, we encounter parentheses, division, addition, and subtraction. According to PEMDAS/BODMAS, we must first address the operations within the parentheses, followed by division, and then addition and subtraction, proceeding from left to right.

Step-by-Step解题：The Art of Unraveling the Equation

Now, let's embark on our step-by-step solution, applying the order of operations to navigate through the equation 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19.

  1. Conquering the Innermost Parentheses:

    Our journey begins with simplifying the innermost parentheses, (9 + 9). This simple addition yields 18. Our equation now transforms into:

    28 - [65 ÷ 5 + 18 - (6 - _)] = 19

  2. Navigating the Remaining Parentheses:

    Next, we turn our attention to the remaining parentheses, (6 - _). However, we encounter a slight hurdle: the presence of the missing value, represented by the underscore. Since we cannot directly evaluate this expression, we will temporarily leave it as is and proceed to the next operation in the order.

  3. The Power of Division:

    According to PEMDAS/BODMAS, division takes precedence over addition and subtraction. Therefore, we perform the division operation, 65 ÷ 5, which results in 13. Our equation now evolves to:

    28 - [13 + 18 - (6 - _)] = 19

  4. Simplifying the Brackets:

    With the division addressed, we now focus on simplifying the expression within the brackets. We perform the addition and subtraction operations from left to right:

    • 13 + 18 = 31
    • 31 - (6 - _)

    Our equation now takes the form:

    28 - [31 - (6 - _)] = 19

  5. Isolating the Unknown:

    Our primary goal is to isolate the missing value. To achieve this, we must first eliminate the brackets and the subtraction operation outside them. We can accomplish this by subtracting 28 from both sides of the equation:

    28 - [31 - (6 - _)] - 28 = 19 - 28

    This simplifies to:

    -[31 - (6 - _)] = -9

  6. Banish the Negative Sign:

    To eliminate the negative sign on the left side, we multiply both sides of the equation by -1:

    -[31 - (6 - _)] * -1 = -9 * -1

    This yields:

    31 - (6 - _) = 9

  7. Isolating the Parentheses (Again):

    We continue our quest to isolate the missing value by subtracting 31 from both sides of the equation:

    31 - (6 - _) - 31 = 9 - 31

    This simplifies to:

    -(6 - _) = -22

  8. The Negative Sign's Return:

    We encounter another negative sign on the left side, which we eliminate by multiplying both sides of the equation by -1:

    -(6 - _) * -1 = -22 * -1

    This results in:

    6 - _ = 22

  9. The Final Revelation: Solving for the Missing Value:

    Finally, we arrive at the решающий step: solving for the missing value. We subtract 6 from both sides of the equation:

    6 - _ - 6 = 22 - 6

    This gives us:

    - _ = 16

    Multiplying both sides by -1, we unveil the missing value:

    _ = -16

Verification: Ensuring Accuracy

To ensure the validity of our solution, we must substitute the value of -16 back into the original equation and verify that it holds true:

28 - [65 ÷ 5 + (9 + 9) - (6 - (-16))] = 19

Following the order of operations:

  • 28 - [65 ÷ 5 + (9 + 9) - (6 + 16)] = 19
  • 28 - [65 ÷ 5 + 18 - 22] = 19
  • 28 - [13 + 18 - 22] = 19
  • 28 - [31 - 22] = 19
  • 28 - 9 = 19
  • 19 = 19

The equation holds true when we substitute -16 for the missing value, confirming the accuracy of our solution.

The End Result: Insights and Key takeaways

In conclusion, by adhering to the order of operations and employing a systematic approach, we successfully solved the equation 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19, revealing the missing value to be -16. This exercise underscores the critical role of PEMDAS/BODMAS in navigating mathematical expressions and the power of methodical problem-solving.

Complex mathematical equations often appear as daunting puzzles, but with a structured approach and a solid understanding of fundamental principles, we can unravel their mysteries. The equation 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19 presents such a challenge, requiring us to navigate a series of operations and ultimately solve for the missing value. In this comprehensive guide, we will dissect this equation step-by-step, emphasizing the importance of the order of operations and demonstrating a methodical approach to problem-solving.

The Foundation: Mastering the Order of Operations

At the heart of solving any mathematical equation lies the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This set of rules serves as our compass, guiding us through the sequence in which operations must be performed to arrive at the correct solution.

PEMDAS/BODMAS dictates that we first address Parentheses/Brackets, followed by Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). This hierarchy ensures consistency and accuracy in our calculations.

In our equation, we encounter a variety of operations: parentheses, division, addition, and subtraction. Adhering to PEMDAS/BODMAS, we must first tackle the operations within parentheses, followed by division, and then proceed with addition and subtraction, moving from left to right.

Step-by-Step Solution: A Journey Through the Equation

Let's embark on our step-by-step journey to solve the equation 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19, meticulously applying the order of operations at each stage.

  1. Unlocking the Innermost Parentheses:

    Our journey commences with simplifying the innermost parentheses, (9 + 9). This straightforward addition yields 18. Our equation now transforms into:

    28 - [65 ÷ 5 + 18 - (6 - _)] = 19

  2. Confronting the Remaining Parentheses:

    Next, we turn our attention to the remaining parentheses, (6 - _). However, we encounter the missing value, represented by the underscore, which prevents us from directly evaluating this expression. We will temporarily set it aside and proceed to the next operation in the order.

  3. The Division Operation:

    According to PEMDAS/BODMAS, division takes precedence over addition and subtraction. Thus, we perform the division operation, 65 ÷ 5, which results in 13. Our equation now evolves to:

    28 - [13 + 18 - (6 - _)] = 19

  4. Simplifying the Brackets:

    With the division addressed, we now focus on simplifying the expression within the brackets. We perform the addition and subtraction operations from left to right:

    • 13 + 18 = 31
    • 31 - (6 - _)

    Our equation now takes the form:

    28 - [31 - (6 - _)] = 19

  5. Isolating the Unknown Value:

    Our primary objective is to isolate the missing value. To achieve this, we must eliminate the brackets and the subtraction operation outside them. We can accomplish this by subtracting 28 from both sides of the equation:

    28 - [31 - (6 - _)] - 28 = 19 - 28

    This simplifies to:

    -[31 - (6 - _)] = -9

  6. Eliminating the Negative Sign:

    To eliminate the negative sign on the left side, we multiply both sides of the equation by -1:

    -[31 - (6 - _)] * -1 = -9 * -1

    This yields:

    31 - (6 - _) = 9

  7. Isolating the Parentheses (Revisited):

    We continue our pursuit of the missing value by subtracting 31 from both sides of the equation:

    31 - (6 - _) - 31 = 9 - 31

    This simplifies to:

    -(6 - _) = -22

  8. The Negative Sign's Encore:

    We encounter another negative sign on the left side, which we eliminate by multiplying both sides of the equation by -1:

    -(6 - _) * -1 = -22 * -1

    This results in:

    6 - _ = 22

  9. Unveiling the Missing Value:

    Finally, we arrive at the decisive step: unveiling the missing value. We subtract 6 from both sides of the equation:

    6 - _ - 6 = 22 - 6

    This gives us:

    - _ = 16

    Multiplying both sides by -1, we reveal the missing value:

    _ = -16

Verification: Ensuring the Solution's Validity

To ensure the validity of our solution, we must substitute the value of -16 back into the original equation and verify that it holds true:

28 - [65 ÷ 5 + (9 + 9) - (6 - (-16))] = 19

Following the order of operations:

  • 28 - [65 ÷ 5 + (9 + 9) - (6 + 16)] = 19
  • 28 - [65 ÷ 5 + 18 - 22] = 19
  • 28 - [13 + 18 - 22] = 19
  • 28 - [31 - 22] = 19
  • 28 - 9 = 19
  • 19 = 19

The equation holds true when we substitute -16 for the missing value, confirming the accuracy of our solution.

Conclusion: Key Takeaways and Insights

In summary, by meticulously adhering to the order of operations and employing a systematic, step-by-step approach, we successfully solved the equation 28 - [65 ÷ 5 + (9 + 9) - (6 - _)] = 19, uncovering the missing value of -16. This exercise underscores the paramount importance of PEMDAS/BODMAS in navigating mathematical expressions and the power of a methodical problem-solving strategy. Furthermore, it highlights the importance of verification to ensure the accuracy of our solutions.