Solving 7 X 2 - 9 + 2 + 14 A Step-by-Step GMDES Guide

In this comprehensive guide, we will walk you through the process of solving the mathematical expression 7 x 2 - 9 + 2 + 14 using the order of operations, commonly remembered by the acronym GMDES (Grouping, Multiplication, Division, Exponent, Addition, Subtraction). Understanding and applying the correct order of operations is crucial for accurately solving mathematical problems. Let’s dive into each step to ensure clarity and precision in our calculation.

Understanding GMDES: The Order of Operations

The acronym GMDES provides a structured approach to solving mathematical expressions, ensuring that operations are performed in the correct sequence. This order is universally accepted and essential for consistent results. GMDES stands for:

• Grouping: Parentheses, brackets, and other grouping symbols are solved first.
• Multiplication: Multiplication operations are performed next.
• Division: Division operations are carried out after multiplication.
• Exponent: Exponents and roots are evaluated after multiplication and division.
• Addition: Addition operations are performed following exponents.
• Subtraction: Subtraction operations are performed last.

By adhering to GMDES, we ensure that complex expressions are simplified systematically, leading to the correct answer. The order prevents ambiguity and ensures that everyone arrives at the same solution, regardless of who is solving the problem. This methodical approach is particularly crucial in more complex mathematical problems, where the sequence of operations can significantly impact the outcome. Let's explore each step in detail as we apply it to our example expression.

Step 1: Identify the Operations

Our expression is 7 x 2 - 9 + 2 + 14. To begin, we need to identify all the mathematical operations present. In this case, we have multiplication, subtraction, and addition. There are no grouping symbols (like parentheses or brackets), exponents, or division operations in this particular expression, which simplifies our initial assessment. Identifying the operations is a critical first step because it sets the stage for applying the GMDES rule correctly. By recognizing each operation, we can prioritize them according to the established order, which will guide us through the subsequent steps. This preliminary step ensures that we don’t miss any operations and that we proceed systematically, minimizing the risk of errors. Proper identification also helps in planning the order in which the operations will be performed, making the entire process more efficient and less prone to mistakes. Let's move on to the next step where we apply the multiplication operation.

Step 2: Perform Multiplication

According to GMDES, multiplication comes before addition and subtraction. Therefore, we start by solving the multiplication part of the expression: 7 x 2. This is a straightforward calculation: 7 multiplied by 2 equals 14. So, the result of this operation is 14. Replacing 7 x 2 with 14 in our original expression, we now have a simplified expression: 14 - 9 + 2 + 14. Performing the multiplication operation first is crucial because it adheres to the established mathematical conventions, ensuring that we arrive at the correct final answer. If we were to perform addition or subtraction before multiplication, we would end up with a different and incorrect result. This step demonstrates the importance of following the GMDES order meticulously. With the multiplication completed, we can now move on to the next set of operations in our expression, which includes addition and subtraction. Let’s continue simplifying the expression by handling these operations in the correct sequence.

Step 3: Perform Addition and Subtraction from Left to Right

Now that we have dealt with the multiplication, we are left with addition and subtraction operations. According to GMDES, addition and subtraction are performed from left to right. This means we tackle the operations in the order they appear in the expression. Our current expression is 14 - 9 + 2 + 14.

First, we perform the subtraction: 14 - 9. This equals 5. So, our expression now becomes 5 + 2 + 14.

Next, we move on to the additions. We add 5 + 2, which equals 7. Our expression is further simplified to 7 + 14.

Finally, we add 7 + 14, which gives us 21. Thus, by performing the addition and subtraction operations from left to right, we have systematically reduced the expression to its simplest form. This left-to-right approach is essential when dealing with operations of the same precedence (like addition and subtraction) to ensure accuracy. The result of this step is a single numerical value, which leads us to the final answer of our original mathematical expression.

Step 4: State the Final Answer

After following the GMDES order of operations, we have simplified the expression 7 x 2 - 9 + 2 + 14 step-by-step. First, we performed the multiplication, then we tackled the addition and subtraction from left to right. Through this process, we have arrived at our final answer. As we calculated in the previous steps, the final result of the expression is 21. This is the solution to our original mathematical problem. By meticulously following the GMDES principles, we have ensured that our calculations are accurate and that we have arrived at the correct answer. Stating the final answer clearly marks the completion of the problem-solving process. It is important to present the solution in a clear and concise manner, so there is no ambiguity about the result. With the final answer of 21, we conclude our step-by-step solution to the expression 7 x 2 - 9 + 2 + 14.

Conclusion

In summary, we have successfully solved the expression 7 x 2 - 9 + 2 + 14 by diligently applying the GMDES order of operations. We began by identifying the operations, then performed the multiplication, and finally, handled the addition and subtraction from left to right. This systematic approach allowed us to arrive at the correct answer of 21. Understanding and using GMDES is crucial for accurate mathematical problem-solving. It provides a clear framework for approaching any mathematical expression, ensuring that operations are performed in the correct sequence. By mastering the GMDES rules, anyone can confidently tackle mathematical problems, regardless of their complexity. This step-by-step guide demonstrates the importance of each step in the GMDES process and reinforces the idea that following a structured approach is key to success in mathematics. Whether you are a student learning the basics or someone looking to refresh your skills, GMDES remains a fundamental concept for mathematical proficiency.