Justification For Step 2 In Solving 12-x=7x+32 Equation

Jul 9, 2025 by ADMIN 56 views

Unlocking the Mystery of Step 2 12-x=7x+32. A Deep Dive into the Justification of Mathematical Steps

In the realm of mathematics, every step in solving an equation is a carefully orchestrated move, justified by fundamental principles and properties. When we delve into the process of solving equations, understanding the justification for each step is crucial for mastering the art of mathematical problem-solving. Let's embark on a journey to dissect the equation 12-x=7x+32, focusing particularly on unraveling the logic behind Step 2. We will explore the properties of equality that govern these transformations and illuminate the path to understanding the correct answer.

Dissecting the Equation: Step-by-Step

Before we zoom in on Step 2, let's first lay out the entire solution process to provide context. This will help us appreciate the role each step plays in the grand scheme of solving for x. The equation we're tackling is:

12 - x = 7x + 32

Now, let's break down the steps involved:

Step 1: -x = 7x + 20

This step is derived by subtracting 12 from both sides of the original equation. The underlying principle here is the subtraction property of equality, which states that if you subtract the same value from both sides of an equation, the equality remains valid. This property is fundamental because it allows us to isolate terms and simplify the equation without altering its balance.

To illustrate, imagine a balance scale. The original equation represents the scale in perfect equilibrium. Subtracting 12 from both sides is like removing equal weights from each side, ensuring the scale remains balanced. Mathematically, this looks like:

12 - x - 12 = 7x + 32 - 12 -x = 7x + 20

This first step effectively moves us closer to isolating the variable x on one side of the equation. It showcases how manipulating equations while adhering to the properties of equality is key to solving for unknowns.

Step 2: -8x = 20

This is the step we're particularly interested in. To get from Step 1 (-x = 7x + 20) to Step 2 (-8x = 20), we need to examine what operation was performed. It's evident that the term 7x has been eliminated from the right side of the equation. This was achieved by subtracting 7x from both sides. Just like in Step 1, we're employing a property of equality, and in this case, it's again the subtraction property of equality. We can also view it as an application of the addition property of equality since subtracting 7x is the same as adding -7x.

Let’s break it down:

Starting with Step 1:

-x = 7x + 20

To eliminate the 7x term from the right side, we subtract 7x from both sides:

-x - 7x = 7x + 20 - 7x

Combining like terms, we get:

-8x = 20

This transformation is pivotal. By subtracting 7x, we've consolidated the x terms on one side of the equation, further simplifying our quest to isolate x. The addition property of equality, when viewed in this light, is a powerful tool for rearranging and simplifying equations while maintaining balance and accuracy.

Step 3: x = -2.5

To arrive at the final solution, we divide both sides of the equation by -8. This step utilizes the division property of equality. This property states that if you divide both sides of an equation by the same non-zero value, the equality holds true. In our case, dividing by -8 isolates x:

-8x / -8 = 20 / -8 x = -2.5

Thus, the solution to the equation is x = -2.5. This final step underscores the importance of isolating the variable to determine its value, and the division property of equality is instrumental in achieving this.

Identifying the Correct Justification for Step 2

Now that we've dissected each step, let's circle back to our main question: What is the justification for Step 2? We've established that the transition from Step 1 (-x = 7x + 20) to Step 2 (-8x = 20) involves subtracting 7x from both sides of the equation. This directly aligns with the addition property of equality. Remember, subtraction can be thought of as adding a negative number, so subtracting 7x is equivalent to adding -7x.

Let's examine the answer choices provided:

A. The division property of equality

This is incorrect. We used division in Step 3, but not in Step 2.

B. The addition property of equality

This is the correct answer. As we've demonstrated, Step 2 is justified by adding -7x to both sides of the equation.

C. The multiplication property of equality

This isn't the primary property at play in Step 2. While multiplying both sides by the same value maintains equality, it's not the direct operation we performed.

D. The subtraction property of equality

This is also the correct answer. As previously mentioned, the subtraction property of equality and the addition property of equality are closely related. Subtracting 7x from both sides is fundamentally the same as adding -7x to both sides.

Therefore, the correct justifications for Step 2 are both the addition property of equality and the subtraction property of equality.

The Significance of Properties of Equality

The properties of equality are the bedrock of algebraic manipulation. They provide the rules that allow us to transform equations while preserving their balance and, ultimately, finding solutions. Without a firm grasp of these properties, solving equations becomes a haphazard process. The addition property of equality (and its close relative, the subtraction property) is particularly vital.

The addition property of equality allows us to add the same value to both sides of an equation without disturbing the equality. This seemingly simple rule is incredibly powerful. It enables us to:

Isolate variables: By strategically adding or subtracting terms, we can move all instances of the variable to one side of the equation.
Simplify equations: Adding or subtracting constants can consolidate terms and make the equation easier to work with.
Maintain balance: The fundamental principle of equation solving is maintaining equality. The addition property ensures that any operation we perform keeps the equation in balance.

Understanding the nuances of the addition property of equality empowers us to approach complex equations with confidence. It transforms the process from a series of guesses to a logical, step-by-step procedure.

Mastering Mathematical Justifications

Being able to solve an equation is one skill, but understanding the why behind each step elevates your mathematical prowess. When you can articulate the justification for every move, you demonstrate a deeper level of comprehension. This not only makes you a more effective problem-solver but also enhances your overall mathematical reasoning.

Here are some strategies for mastering mathematical justifications:

Identify the operation: What action was performed to transform the equation from one step to the next? Was something added, subtracted, multiplied, or divided?
Connect to a property: Which property of equality (or other mathematical principle) justifies the operation? Review the core properties and understand their implications.
Articulate the reasoning: Explain in your own words why the property applies and how it maintains the equality.
Practice consistently: The more you practice solving equations and justifying your steps, the more natural the process will become.

By actively engaging with the justifications behind mathematical steps, you'll develop a more profound understanding of the subject and become a more confident and capable mathematician. You'll move beyond rote memorization and truly grasp the underlying principles that govern mathematical operations.

Conclusion: The Power of Understanding the "Why"

In conclusion, the justification for Step 2 in the solution process of the equation 12-x=7x+32 lies in the addition property of equality and subtraction property of equality. By subtracting 7x from both sides (or adding -7x), we maintain the balance of the equation while moving closer to isolating the variable x. This exercise highlights the critical importance of understanding the properties of equality in solving equations.

As we've explored, each step in solving an equation is a deliberate action rooted in mathematical principles. By dissecting these steps and understanding their justifications, we not only arrive at the correct answer but also cultivate a deeper appreciation for the elegance and logic of mathematics. Mastering these justifications is a cornerstone of mathematical fluency, empowering us to tackle increasingly complex problems with clarity and confidence. So, the next time you solve an equation, don't just focus on the how; take the time to understand the why. This deeper understanding will unlock a new level of mathematical mastery.

By embracing the justifications, we transform mathematics from a set of rules to a logical and interconnected system. This, in turn, empowers us to approach new challenges with a sense of understanding and control. The journey to mathematical proficiency is not just about finding answers; it's about understanding the fundamental principles that underpin the entire process.

By taking the time to analyze each step and its justification, we not only solve the problem at hand but also strengthen our mathematical foundation for future endeavors. So, let us continue to explore, question, and justify, unlocking the full potential of our mathematical minds.