Paula's Equation Solving Steps Identifying Division Property Application

This article delves into the step-by-step solution of an equation presented by Paula, focusing on pinpointing where she applied the division property of equality. Understanding this property is crucial in algebra, as it allows us to isolate variables and solve for their values. We will meticulously examine each step of Paula's solution, ensuring clarity and comprehension for readers of all levels. Our goal is not just to identify the step where the division property was used, but also to explain why it was necessary and how it contributes to finding the solution. This approach will enhance the reader's understanding of algebraic problem-solving strategies and the importance of each property used.

H2: Understanding Paula's Steps

Paula's solution involves a series of algebraic manipulations designed to isolate the variable 'x'. Let's break down each step to clearly understand her process and pinpoint where the division property comes into play. Her initial equation and subsequent steps are as follows:

Step 1: The journey begins with the equation $-4(x+8)-2x=25$. This is the starting point, the initial state of our puzzle. In this step, we observe a combination of terms involving the variable 'x' and constants. The left side of the equation presents an expression that needs simplification before we can isolate 'x'. The presence of parentheses indicates the need for the distributive property, a key step in unraveling this equation. This initial step sets the stage for the subsequent algebraic manipulations that will lead us to the solution.

Step 2: Here, Paula applies the distributive property, expanding the term $-4(x+8)$ to $-4x-32$. The equation now reads $-4x-32-2x=25$. The distributive property is a fundamental concept in algebra, allowing us to eliminate parentheses and combine like terms. By multiplying -4 by both 'x' and '8', Paula has successfully removed the parentheses, making the equation easier to work with. This step is a critical bridge between the initial equation and the simplified forms that will follow. It demonstrates a solid understanding of algebraic principles and sets the foundation for further simplification.

Step 3: Next, Paula combines like terms, simplifying $-4x-2x$ to $-6x$. The equation is now $-6x-32=25$. Combining like terms is a standard algebraic technique used to simplify expressions and equations. By identifying and combining the 'x' terms, Paula has reduced the complexity of the left side of the equation. This step streamlines the equation, bringing us closer to isolating the variable 'x'. It showcases a practical application of algebraic principles aimed at making the equation more manageable and solvable.

Step 4: Paula isolates the term with 'x' by adding 32 to both sides, resulting in $-6x=57$. Isolating the variable term is a crucial step in solving equations. By adding 32 to both sides, Paula has effectively moved the constant term from the left side to the right side of the equation. This manipulation maintains the balance of the equation while bringing us closer to isolating 'x'. This step is a clear demonstration of the addition property of equality, a fundamental principle in algebraic problem-solving.

Step 5: Finally, Paula divides both sides by -6 to solve for 'x', obtaining x=-9 rac{1}{2}. This is the culmination of the algebraic manipulations, where the variable 'x' is finally isolated. By dividing both sides of the equation by -6, Paula has applied the division property of equality. This property is the key to unlocking the value of 'x'. The result, x=-9 rac{1}{2}, represents the solution to the original equation. This final step showcases the power of algebraic principles in solving for unknown variables.

H2: Identifying the Division Property

The division property of equality states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. Paula utilized this property between Step 4 and Step 5. In Step 4, she had the equation $-6x = 57$. To isolate 'x', she divided both sides by -6, which is the division property in action. This step is crucial because it directly leads to the solution for 'x'. Without this step, the variable would remain multiplied by -6, and we wouldn't know its individual value.

The division property of equality is one of the fundamental principles in algebra, ensuring that equations remain balanced during manipulations. It's a cornerstone of solving for variables and understanding algebraic relationships. In Paula's case, this property was the final piece of the puzzle, allowing her to determine the value of 'x' and complete the solution.

Understanding the division property, and when to apply it, is paramount for anyone studying algebra. It's not just about performing a calculation; it's about understanding the underlying mathematical principles that govern equation solving. Paula's solution provides a clear example of how this property is used in practice, highlighting its importance in the broader context of algebraic problem-solving.

H2: Why the Division Property is Important

The division property, as demonstrated in Paula's solution, is not just a mathematical rule; it's a critical tool in the algebraic arsenal. It allows us to undo multiplication and isolate variables, which is the fundamental goal of solving equations. Imagine trying to find the value of 'x' in the equation $-6x = 57$ without the division property. It would be like trying to unscramble an egg – nearly impossible! The division property provides the method to systematically peel away the layers surrounding the variable until it stands alone, revealing its true value.

Furthermore, the division property ensures that our algebraic manipulations are valid and maintain the equality. This principle is crucial for the integrity of the solution. When we divide both sides of an equation by the same number, we are essentially performing the same operation on both sides, preserving the balance. This balance is the essence of an equation – the left side must always equal the right side. Without this balance, the solution would be meaningless.

The importance of the division property extends beyond simple equations. It's a building block for more advanced algebraic concepts and techniques. From solving systems of equations to manipulating formulas in physics and engineering, the division property is a recurring theme. Mastering this property is not just about solving a single equation; it's about developing a deeper understanding of how mathematical relationships work.

In Paula's solution, the division property was the linchpin that connected the simplified equation to the final answer. It transformed $-6x = 57$ into the clear and concise solution x = -9 rac{1}{2}. This transformation highlights the power and elegance of the division property in algebraic problem-solving. It's a testament to the importance of understanding and applying fundamental mathematical principles.

H2: Common Mistakes and How to Avoid Them

When applying the division property, several common mistakes can lead to incorrect solutions. Recognizing these pitfalls and understanding how to avoid them is crucial for mastering this essential algebraic skill. One of the most frequent errors is dividing only one side of the equation. Remember, the division property, like all equality properties, requires performing the operation on both sides to maintain balance. Failing to do so will disrupt the equation's integrity and lead to a wrong answer. For example, in Paula's case, dividing only $-6x$ by -6 would not have solved for 'x'.

Another common mistake is dividing by zero. Division by zero is undefined in mathematics and will render an equation meaningless. This error often arises from a misunderstanding of the properties of zero or a simple oversight. Always double-check that the number you are dividing by is not zero. If it is, consider alternative approaches to solving the equation.

Sign errors are another potential trap when applying the division property. Remember that dividing a positive number by a negative number results in a negative quotient, and vice versa. Pay close attention to the signs of the numbers involved to avoid making careless errors. In Paula's solution, dividing 57 by -6 required careful attention to the negative sign to arrive at the correct answer.

To avoid these mistakes, it's essential to practice applying the division property methodically and consistently. Always double-check your work, paying close attention to the balance of the equation and the signs of the numbers. If you encounter a mistake, try to understand why it occurred so you can avoid it in the future. With careful practice and attention to detail, you can master the division property and confidently solve a wide range of algebraic equations.

H2: Conclusion

In conclusion, Paula's solution vividly illustrates the application of the division property of equality. This property, employed between Step 4 and Step 5, is a cornerstone of algebraic problem-solving. By dividing both sides of the equation by -6, Paula successfully isolated 'x' and determined its value. Understanding the division property, along with its importance in maintaining equation balance, is crucial for anyone venturing into the world of algebra. Avoiding common mistakes, such as dividing by zero or neglecting signs, further ensures accurate solutions. As we've seen, the division property is not just a mathematical rule; it's a powerful tool that unlocks the solutions to countless algebraic problems. Mastering this property is a significant step towards achieving algebraic fluency.

In this article, we have not only identified the step where Paula used the division property but also explored its significance and potential pitfalls. By understanding the underlying principles and practicing diligently, readers can confidently apply the division property and tackle algebraic challenges with ease. The journey through Paula's solution serves as a valuable lesson in the power and precision of algebraic manipulations.