Maria's Equation Solving Methods And Order Of Operations

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Introduction

In this article, we will delve into the step-by-step process Maria used to solve the algebraic equation 4(3x−8)−4=24{\begin{aligned} 4(3x - 8) - 4 = 24 \end{aligned}}. We will meticulously examine each step, identifying the mathematical principles and methods applied. Our goal is to provide a clear and comprehensive understanding of the solution, making it accessible to anyone interested in algebra. This exploration will not only showcase Maria's problem-solving skills but also serve as a valuable learning resource for mastering algebraic techniques. Algebraic equations are fundamental to mathematics, and understanding how to solve them is crucial for various fields, including science, engineering, and economics. Maria's approach highlights the importance of following a logical and systematic method to arrive at the correct solution. By breaking down the equation into manageable steps, she demonstrates the power of applying fundamental algebraic principles. This article will analyze these principles, making it easier for readers to grasp the underlying concepts and apply them to their own problem-solving endeavors. The beauty of mathematics lies in its precision and consistency, and Maria's solution exemplifies these qualities. Each step she takes is justified by a specific mathematical rule, ensuring the accuracy of the final answer. This methodical approach is what makes algebra so powerful and reliable. We aim to unravel this process, providing a clear pathway for understanding and replicating Maria's success in solving algebraic equations. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this article will provide valuable insights and practical guidance. Join us as we embark on this mathematical journey, dissecting Maria's method and uncovering the logic behind each step. By the end of this article, you will have a solid understanding of the techniques used and be better equipped to tackle similar algebraic challenges.

Step-by-Step Solution Analysis

Maria began with the equation:

4(3x−8)−4=24{\begin{aligned} 4(3x - 8) - 4 = 24 \end{aligned}}

The first method Maria used was the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. Maria applied this property to the term 4(3x - 8). By distributing the 4 across the terms inside the parentheses, she multiplied 4 by 3x and 4 by -8. This gave her 12x - 32. The equation then became:

12x−32−4=24{\begin{aligned} 12x - 32 - 4 = 24 \end{aligned}}

The next method Maria employed was combining like terms. Like terms are terms that have the same variable raised to the same power. In this equation, -32 and -4 are like terms because they are both constants. Maria combined these terms by adding them together: -32 + (-4) = -36. This simplified the equation to:

12x−36=24{\begin{aligned} 12x - 36 = 24 \end{aligned}}

To isolate the term with the variable (12x), Maria added 36 to both sides of the equation. This is based on the addition property of equality, which states that if you add the same number to both sides of an equation, the equation remains balanced. Adding 36 to both sides, the equation became:

12x−36+36=24+36{\begin{aligned} 12x - 36 + 36 = 24 + 36 \end{aligned}}

12x=60{\begin{aligned} 12x = 60 \end{aligned}}

Finally, to solve for x, Maria divided both sides of the equation by 12. This is based on the division property of equality, which states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. Dividing both sides by 12, she got:

12x12=6012{\begin{aligned} \frac{12x}{12} = \frac{60}{12} \end{aligned}}

x=5{\begin{aligned} x = 5 \end{aligned}}

Thus, Maria found the solution to the equation to be x = 5. Maria's step-by-step approach to solving this equation demonstrates a solid understanding of algebraic principles. She skillfully applied the distributive property to eliminate parentheses, combined like terms to simplify the equation, and used the addition and division properties of equality to isolate the variable. Each step was executed with precision, leading to the correct solution. This methodical approach is a hallmark of effective problem-solving in algebra, and it serves as a valuable lesson for anyone learning to solve equations. The importance of understanding these fundamental algebraic techniques cannot be overstated, as they form the basis for more advanced mathematical concepts. Maria's ability to apply these techniques correctly highlights the significance of practice and a clear understanding of the underlying principles. By breaking down the equation into manageable steps, she demonstrates how even complex problems can be solved through the systematic application of basic rules. This approach not only ensures accuracy but also enhances comprehension, making it easier to tackle similar problems in the future. The process Maria followed is a testament to the power of algebraic manipulation and the importance of adhering to the properties of equality. Her solution serves as a model for students and anyone seeking to improve their algebraic skills. By following her method, one can develop a strong foundation in equation solving and gain confidence in tackling mathematical challenges. The clarity and precision of her solution underscore the elegance of algebra as a problem-solving tool. The step-by-step approach not only yields the correct answer but also provides a clear and logical pathway that is easy to follow and understand.

Methods Used and Their Order

In summary, Maria used the following methods in the following order:

  1. Distributive Property: To eliminate the parentheses by multiplying 4 by both terms inside the parentheses (3x and -8).
  2. Combining Like Terms: To simplify the equation by adding the constant terms (-32 and -4).
  3. Addition Property of Equality: To isolate the term with the variable by adding 36 to both sides of the equation.
  4. Division Property of Equality: To solve for x by dividing both sides of the equation by 12.

This sequence of steps is crucial for solving the equation correctly. Applying the distributive property first simplifies the equation, making it easier to work with. Combining like terms further simplifies the equation, reducing the number of terms and making it more manageable. The addition and division properties of equality are then used to isolate the variable and solve for its value. Each step builds upon the previous one, ensuring a logical and systematic progression toward the solution. The order in which these methods are applied is not arbitrary; it is carefully chosen to facilitate the solution process. For instance, attempting to combine like terms before applying the distributive property would be more complex and could lead to errors. Similarly, isolating the variable before simplifying the equation would make the process more cumbersome. Maria's methodical approach demonstrates an understanding of the underlying principles of algebra and the importance of applying them in the correct order. This methodical approach is a key to success in algebra, and it is a skill that can be developed through practice and a clear understanding of the rules. By following a structured approach, one can avoid common errors and arrive at the correct solution more efficiently. The sequence of steps Maria used is a testament to the power of algebraic manipulation when applied in a logical and systematic manner. Each step is justified by a specific mathematical property, ensuring the accuracy of the solution. This clarity and precision are hallmarks of effective problem-solving in algebra. By understanding the order in which these methods are applied, one can gain a deeper appreciation for the structure and logic of algebraic equations. The sequential nature of the solution process highlights the interconnectedness of the different algebraic principles. Each step relies on the previous one, and together they form a coherent and effective method for solving equations. This understanding is essential for developing proficiency in algebra and for tackling more complex mathematical problems.

Conclusion

Maria's solution to the equation highlights the importance of a systematic approach in algebra. By correctly applying the distributive property, combining like terms, and using the addition and division properties of equality in the right order, she efficiently found the value of x. This example serves as a valuable lesson in algebraic problem-solving, emphasizing the significance of understanding the underlying principles and applying them methodically. The process Maria used is not only a means to an end but also a demonstration of the elegance and logic of mathematics. Each step is justified by a specific rule, and the sequence of steps is carefully chosen to simplify the equation and isolate the variable. This clarity and precision are what make algebra such a powerful tool for solving problems in a variety of fields. By mastering these fundamental techniques, one can gain confidence in their ability to tackle mathematical challenges and develop a deeper appreciation for the beauty of algebraic reasoning. The lessons learned from Maria's solution extend beyond the specific equation she solved; they provide a framework for approaching any algebraic problem. The key is to break the problem down into manageable steps, identify the relevant principles, and apply them in a logical order. This approach not only leads to the correct answer but also fosters a deeper understanding of the underlying concepts. Maria's solution is a testament to the power of careful analysis and systematic problem-solving. It underscores the importance of not just knowing the rules but also understanding how and when to apply them. This understanding is what distinguishes a proficient problem-solver from someone who simply memorizes formulas. By studying Maria's method, students and learners can gain valuable insights into the art of algebraic problem-solving and develop the skills necessary to succeed in mathematics. The methodical approach she used is a valuable model for anyone seeking to improve their algebraic skills. It demonstrates the importance of planning, precision, and a clear understanding of the rules. By emulating Maria's approach, one can develop a strong foundation in algebra and gain the confidence to tackle more complex mathematical challenges. The principles she applied are universal and can be applied to a wide range of problems, making her solution a valuable resource for anyone learning algebra.