What Properties Of Equality Did Kira Use To Solve The Equation $\frac{7}{2}-2 X=\frac{11}{2}$?

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In this article, we will delve into Kira's method for solving the equation 72−2x=112\frac{7}{2}-2x = \frac{11}{2}. We will meticulously examine the steps Kira took and elucidate the properties of equality that she employed to arrive at the solution. Understanding these properties is crucial for mastering algebraic manipulations and solving a wide array of mathematical problems. This exploration will not only help clarify Kira's approach but also solidify your grasp of fundamental algebraic principles.

Decoding Kira's Steps

To truly appreciate Kira's solution, let's dissect each step with precision. The initial equation presented is 72−2x=112\frac{7}{2}-2x = \frac{11}{2}. Kira's subsequent actions reveal her understanding of how to isolate the variable x and ultimately determine its value. The process involves strategically applying properties of equality to maintain the balance of the equation while simplifying it.

The first step Kira took appears to be an attempt to isolate the term containing x. She seems to have subtracted 72\frac{7}{2} from both sides of the equation. This is a critical move in solving for x because it begins the process of getting the x term by itself on one side of the equation. Let's analyze this manipulation more closely to ensure we understand the underlying principle.

The Subtraction Property of Equality

The key property at play here is the Subtraction Property of Equality. This property is a cornerstone of algebraic problem-solving. It asserts that if you subtract the same value from both sides of an equation, the equation remains balanced. In simpler terms, if a = b, then a - c = b - c. This might sound straightforward, but it's a powerful tool for manipulating equations.

In Kira's case, she recognized that subtracting 72\frac{7}{2} from both sides would help isolate the term with x. This strategic move demonstrates an understanding of how to use the Subtraction Property of Equality to simplify the equation. The application of this property is evident in the transition from the original equation to the subsequent step, where the constant term on the left side begins to disappear. This is a typical and valid algebraic technique used to isolate the variable.

Identifying the Error

However, there's a subtle error in how Kira wrote out this step. While the idea of subtracting 72\frac{7}{2} from both sides is correct, the notation used is slightly confusing and ultimately leads to an incorrect intermediate expression. The way Kira wrote it, "72=72−2x=112−72\frac{7}{2}=\frac{7}{2}-2 x=\frac{11}{2}-\frac{7}{2}", suggests a chain of equalities that isn't quite accurate. This notation implies that 72\frac{7}{2} is equal to the expression 72−2x\frac{7}{2}-2x, which is not true based on the original equation. The proper application of the Subtraction Property should result in a clearer and more accurate representation of the equation's transformation.

The Correct Application

To correctly apply the Subtraction Property, Kira should have shown the subtraction more explicitly. The step should have looked more like this:

72−2x−72=112−72\frac{7}{2} - 2x - \frac{7}{2} = \frac{11}{2} - \frac{7}{2}

This clearer notation makes it immediately evident that 72\frac{7}{2} is being subtracted from both sides of the equation. It avoids the confusing chain of equalities and directly illustrates the application of the Subtraction Property.

By understanding the correct application of the Subtraction Property, we can not only identify the subtle error in Kira's notation but also reinforce the importance of clear and accurate mathematical communication.

Isolating the Variable: A Step-by-Step Analysis

After the initial attempt to subtract 72\frac{7}{2}, Kira's next line presents us with -2x = -2 = \frac{4}{12}. This step consolidates the result of the previous operation and sets the stage for isolating x completely. Let's dissect this line, paying close attention to the arithmetic and the properties that are implicitly being used.

The first part, -2x = -2, seems to be the result of simplifying the equation after subtracting 72\frac{7}{2} from both sides. Let's verify this:

72−2x−72=112−72\frac{7}{2} - 2x - \frac{7}{2} = \frac{11}{2} - \frac{7}{2}

Simplifying the left side, the 72\frac{7}{2} terms cancel out, leaving us with -2x. On the right side, we have 112−72\frac{11}{2} - \frac{7}{2}, which simplifies to 42\frac{4}{2}, and further reduces to 2. However, there's a subtle error here! The result should be positive 2, not -2. Kira seems to have made a sign error in the subtraction. This highlights the importance of careful attention to detail when performing arithmetic operations in algebraic manipulations.

The Division Property of Equality

To isolate x completely, we need to address the coefficient -2 that is multiplying it. This is where the Division Property of Equality comes into play. This property states that if you divide both sides of an equation by the same non-zero value, the equation remains balanced. Mathematically, if a = b, then a/c = b/c, provided that c is not zero.

In our case, we need to divide both sides of the equation -2x = 2 (the corrected version) by -2. This will isolate x on the left side and give us its value on the right side.

Identifying the Second Error

However, let's look at the second part of Kira's equation: -2 = \frac{4}{12}. This is clearly incorrect. The number -2 is not equal to the fraction 412\frac{4}{12}. This indicates a misunderstanding of basic arithmetic or a simple calculation mistake. This error further emphasizes the need for meticulousness and careful checking of each step in the problem-solving process.

Even if the previous steps had been correct, this error would lead to a completely wrong answer. It's essential to recognize such discrepancies and correct them to ensure the final solution is accurate.

The Correct Application of Division

To correctly isolate x, we should perform the division as follows:

−2x−2=2−2\frac{-2x}{-2} = \frac{2}{-2}

This simplifies to:

x = -1

This is the correct value of x that satisfies the original equation. By carefully applying the Division Property of Equality and avoiding arithmetic errors, we arrive at the accurate solution. This process highlights the importance of both understanding the properties of equality and executing the arithmetic operations flawlessly.

Properties of Equality: The Foundation of Algebra

Kira's attempt, despite the errors, provides a valuable opportunity to discuss the fundamental Properties of Equality. These properties are the bedrock of algebraic manipulation, allowing us to transform equations while preserving their balance and validity. Mastering these properties is crucial for success in algebra and beyond.

A Comprehensive Overview

Let's take a moment to review the key Properties of Equality:

  1. Addition Property of Equality: If a = b, then a + c = b + c. Adding the same value to both sides of an equation maintains the equality.
  2. Subtraction Property of Equality: If a = b, then a - c = b - c. Subtracting the same value from both sides of an equation maintains the equality. As discussed earlier, Kira attempted to use this property, but her notation was slightly off, and she made a sign error in the subsequent simplification.
  3. Multiplication Property of Equality: If a = b, then a * c = b * c. Multiplying both sides of an equation by the same value maintains the equality. This property is particularly useful when dealing with fractions or reciprocals.
  4. Division Property of Equality: If a = b, then a / c = b / c (where c ≠ 0). Dividing both sides of an equation by the same non-zero value maintains the equality. Kira correctly identified this property as necessary to isolate x, but arithmetic errors hindered her progress.
  5. Reflexive Property of Equality: a = a. This property states that any value is equal to itself. While seemingly obvious, it's a fundamental concept in mathematical reasoning.
  6. Symmetric Property of Equality: If a = b, then b = a. This property allows us to reverse the sides of an equation without changing its meaning. It's a simple but useful property for rearranging equations to a more convenient form.
  7. Transitive Property of Equality: If a = b and b = c, then a = c. This property allows us to connect equalities. If two values are equal to the same value, then they are equal to each other.
  8. Substitution Property of Equality: If a = b, then a can be substituted for b in any equation or expression. This property is frequently used in simplifying expressions and solving systems of equations.

Applying the Properties in Concert

These properties are not used in isolation; rather, they are often applied in combination to solve equations. In Kira's example, the Subtraction Property of Equality and the Division Property of Equality were both necessary to isolate x. The strategic application of these properties is what allows us to manipulate equations and find solutions.

The Importance of Accuracy

While understanding the Properties of Equality is crucial, accuracy in arithmetic is equally important. Kira's errors highlight that even with a grasp of the underlying principles, mistakes in calculations can lead to incorrect results. Therefore, it's essential to double-check each step and ensure that all arithmetic operations are performed correctly.

By mastering the Properties of Equality and cultivating meticulousness in calculations, you'll be well-equipped to tackle a wide range of algebraic problems with confidence and accuracy. This understanding forms a solid foundation for more advanced mathematical concepts.

Correcting Kira's Solution: A Step-by-Step Approach

To solidify our understanding, let's rework Kira's solution from the beginning, applying the Properties of Equality correctly and avoiding the arithmetic errors she made. This step-by-step approach will not only lead us to the correct answer but also reinforce the proper techniques for solving equations.

Starting with the Original Equation

We begin with the original equation:

72−2x=112\frac{7}{2} - 2x = \frac{11}{2}

Applying the Subtraction Property of Equality

Our goal is to isolate the term with x, so we'll use the Subtraction Property of Equality to subtract 72\frac{7}{2} from both sides:

72−2x−72=112−72\frac{7}{2} - 2x - \frac{7}{2} = \frac{11}{2} - \frac{7}{2}

Simplifying both sides, we get:

-2x = \frac{4}{2}

Further simplifying the right side:

-2x = 2

Applying the Division Property of Equality

Now, to isolate x, we'll use the Division Property of Equality and divide both sides by -2:

−2x−2=2−2\frac{-2x}{-2} = \frac{2}{-2}

This simplifies to:

x = -1

Verifying the Solution

To ensure our solution is correct, we can substitute x = -1 back into the original equation:

72−2(−1)=112\frac{7}{2} - 2(-1) = \frac{11}{2}

Simplifying:

72+2=112\frac{7}{2} + 2 = \frac{11}{2}

Converting 2 to a fraction with a denominator of 2:

72+42=112\frac{7}{2} + \frac{4}{2} = \frac{11}{2}

112=112\frac{11}{2} = \frac{11}{2}

Since the equation holds true, our solution x = -1 is correct.

The Importance of Verification

This verification step is crucial. It allows us to catch any errors that may have occurred during the solving process. By substituting the solution back into the original equation, we can be confident that our answer is accurate.

Key Takeaways

This step-by-step correction of Kira's solution highlights several important points:

  • Clear Notation: Using clear and accurate notation is essential for avoiding confusion and errors.
  • Careful Arithmetic: Paying close attention to detail and performing arithmetic operations meticulously is crucial.
  • Strategic Application of Properties: Understanding the Properties of Equality and applying them strategically is key to solving equations.
  • Verification: Always verify your solution by substituting it back into the original equation.

By following these guidelines, you can confidently solve algebraic equations and avoid common pitfalls.

Conclusion: Mastering Equations Through Understanding

In conclusion, Kira's attempt to solve the equation 72−2x=112\frac{7}{2}-2x = \frac{11}{2} provides a valuable learning opportunity. While her approach demonstrated an understanding of the fundamental Properties of Equality, errors in notation and arithmetic led to an incorrect solution. By dissecting her steps, identifying the mistakes, and reworking the problem correctly, we've reinforced the importance of clear communication, meticulous calculations, and the strategic application of algebraic principles.

The Power of Properties of Equality

We've emphasized the central role of the Properties of Equality in manipulating equations and isolating variables. These properties are not mere rules to be memorized; they are the foundation upon which algebraic problem-solving is built. Understanding why these properties work is just as important as knowing how to apply them.

The Significance of Accuracy and Verification

Furthermore, we've highlighted the critical need for accuracy in arithmetic. Even a minor error can derail the entire solution process. This underscores the importance of double-checking each step and ensuring that all calculations are correct. The verification step, where the solution is substituted back into the original equation, serves as a final safeguard against errors.

Building a Solid Foundation

By mastering the Properties of Equality, cultivating careful calculation habits, and consistently verifying solutions, you can build a solid foundation in algebra. This foundation will empower you to tackle more complex problems and excel in your mathematical studies. Remember, mathematics is not just about finding the right answer; it's about understanding the process and developing the skills to solve problems effectively and accurately.

Encouragement for Continued Learning

We encourage you to continue practicing and exploring algebraic concepts. The more you work with equations, the more comfortable and confident you will become. Embrace challenges as opportunities to learn and grow, and never hesitate to seek clarification or assistance when needed. With dedication and perseverance, you can unlock the power of algebra and its many applications.

This exploration of Kira's equation-solving method serves as a testament to the value of understanding the underlying principles, paying attention to detail, and embracing the iterative process of learning in mathematics. By applying these lessons, you can confidently navigate the world of equations and beyond.