Identifying The Algebraic Property Used In Equation Solving
In the realm of mathematics, solving equations is akin to navigating a labyrinth, each step meticulously chosen to unravel the unknown. Simon's journey through this algebraic maze, as depicted in the equation-solving steps, presents an intriguing puzzle. Our task is to dissect his methodology, pinpointing the precise property that propelled him from step 3 to step 4. To embark on this quest, let us first immerse ourselves in the intricacies of Simon's calculations.
Simon's Step-by-Step Equation Solving
Simon's expedition commences with the equation:
Step 1: -6(x + 7) = 2(x - 11)
The initial equation serves as the bedrock upon which the subsequent steps are built. It introduces the variable 'x' and establishes a relationship that must be maintained throughout the solution process. The equation's structure hints at the utilization of the distributive property, a mathematical principle that dictates how expressions within parentheses are handled.
Step 2: -6x - 42 = 2x - 22
In this step, Simon masterfully employs the distributive property. He multiplies -6 by both 'x' and '7' on the left side of the equation, yielding -6x - 42. Similarly, he multiplies 2 by both 'x' and '-11' on the right side, resulting in 2x - 22. This step effectively eliminates the parentheses, paving the way for further simplification.
Step 3: -8x - 42 = -22
Here, Simon executes a strategic maneuver to consolidate like terms. He subtracts 2x from both sides of the equation. This operation serves to gather the 'x' terms on one side, a crucial step in isolating the variable. The equation now stands as -8x - 42 = -22, a testament to Simon's algebraic prowess.
Step 4: -8x = 20
This is the pivotal step, the crux of our inquiry. Simon has transitioned from -8x - 42 = -22 to -8x = 20. The burning question is: Which property empowered this transformation? To decipher this, we must delve into the fundamental properties of equality.
The Guiding Light: Properties of Equality
The properties of equality are the bedrock of algebraic manipulation. They are the unwritten rules that govern how we can transform equations without altering their fundamental truth. Among these properties, the addition property of equality shines as the most likely candidate in Simon's maneuver.
Addition Property of Equality: The Key Unlocked
The addition property of equality states that adding the same value to both sides of an equation preserves the equality. In simpler terms, if we have an equation A = B, then A + C = B + C. This seemingly simple principle is a cornerstone of equation solving.
In Simon's case, we observe that he has effectively eliminated the -42 term from the left side of the equation. This feat is accomplished by adding 42 to both sides of the equation:
-8x - 42 + 42 = -22 + 42
This operation neatly cancels out the -42 on the left, leaving us with:
-8x = 20
Thus, the addition property of equality emerges as the hero of this step, the property that Simon implicitly invoked to propel his solution forward.
Distinguishing the Properties: A Necessary Detour
While the addition property of equality takes center stage here, it is prudent to briefly touch upon other properties that might initially seem relevant. These include the subtraction, multiplication, and division properties of equality. However, a closer examination reveals that these properties do not directly explain Simon's transition from step 3 to step 4.
- The subtraction property of equality is the inverse of the addition property, where subtracting the same value from both sides maintains equality.
- The multiplication property of equality dictates that multiplying both sides of an equation by the same non-zero value preserves equality.
- The division property of equality, conversely, states that dividing both sides of an equation by the same non-zero value maintains equality.
In Simon's step, we observe an addition operation at play, making the addition property of equality the most fitting explanation.
Conclusion: The Addition Property of Equality Triumphs
Simon's equation-solving journey provides a captivating glimpse into the world of algebraic manipulation. By meticulously dissecting his steps, we have unveiled the property that enabled his transformation from step 3 to step 4: the addition property of equality.
This property, a fundamental pillar of equation solving, allows us to add the same value to both sides of an equation without disrupting its balance. It is through this principle that Simon effectively eliminated the constant term, paving the way for the isolation of the variable 'x'.
In the grand tapestry of mathematics, each property plays a unique role, and the addition property of equality stands as a testament to the power of maintaining balance in the pursuit of solutions. Simon's journey serves as a reminder that understanding these properties is not just about memorizing rules; it's about wielding the tools that unlock the secrets hidden within equations.
When it comes to solving equations, each step is carefully chosen to bring us closer to the solution. In this article, we will dissect a step-by-step equation-solving process and pinpoint the exact property used in a crucial transformation. Understanding these properties is essential for mastering algebra and solving more complex problems. We'll explore the specific steps taken, the equation's evolution, and ultimately, identify the key property that made the transition possible. This journey through algebraic manipulation will not only reveal the answer but also solidify your understanding of equation-solving principles.
Step-by-Step Breakdown of the Equation Solving Process
The problem we're tackling involves a series of steps taken to solve for the variable 'x.' Let's revisit the equation-solving steps presented:
Step 1: -6(x + 7) = 2(x - 11)
This is the initial equation. It contains parentheses and requires simplification before we can isolate 'x.' This first step sets the stage for the subsequent transformations and hints at the application of the distributive property to eliminate the parentheses.
Step 2: -6x - 42 = 2x - 22
In this step, the distributive property is applied. On the left side, -6 is multiplied by both 'x' and '7,' resulting in -6x - 42. Similarly, on the right side, 2 is multiplied by both 'x' and '-11,' yielding 2x - 22. This step successfully expands the equation and prepares it for further simplification by combining like terms.
Step 3: -8x - 42 = -22
Here, the goal is to consolidate the 'x' terms. This is achieved by subtracting 2x from both sides of the equation. This strategic move groups the 'x' terms together, making the equation easier to manipulate and ultimately solve for 'x.'
Step 4: -8x = 20
This is the crucial step we need to analyze. The question asks which property was used to get from Step 3 to Step 4. Our task is to identify the specific algebraic manipulation that transforms the equation from -8x - 42 = -22 to -8x = 20. To do this, we need to consider the properties of equality and how they apply to equation solving.
Identifying the Key Property
The transition from Step 3 to Step 4 involves a specific algebraic manipulation. Let's focus on the change that occurred: the -42 term on the left side of the equation disappeared. This strongly suggests the use of the addition property of equality.
The Power of the Addition Property of Equality
The addition property of equality is a fundamental principle in algebra. It states that if you add the same value to both sides of an equation, the equation remains balanced. In other words, if A = B, then A + C = B + C.
To understand how this applies to our problem, let's examine Step 3 and Step 4 closely:
Step 3: -8x - 42 = -22
Step 4: -8x = 20
Notice that to get from Step 3 to Step 4, we need to eliminate the -42 term on the left side. This can be achieved by adding 42 to both sides of the equation:
-8x - 42 + 42 = -22 + 42
This simplifies to:
-8x = 20
Therefore, the property used to get from Step 3 to Step 4 is indeed the addition property of equality. By adding 42 to both sides, we successfully isolated the term with 'x' and moved closer to solving for the variable.
Why Not Other Properties?
It's important to consider why other properties of equality are not the correct answer in this case. Let's briefly discuss a few:
- Subtraction Property of Equality: This property involves subtracting the same value from both sides, which is the opposite of what occurred in this step.
- Multiplication Property of Equality: This property involves multiplying both sides by the same value. There was no multiplication performed to get from Step 3 to Step 4.
- Division Property of Equality: This property involves dividing both sides by the same value. Again, no division was used in this step.
The addition property of equality is the only property that accurately describes the transformation between Step 3 and Step 4.
The Addition Property of Equality The Cornerstone of Equation Solving
In conclusion, the property used to get from Step 3 to Step 4 in the given equation-solving process is the addition property of equality. This property is a cornerstone of algebraic manipulation, allowing us to add the same value to both sides of an equation without changing its balance. By understanding and applying these properties, we can effectively solve equations and unlock the solutions they hold.
This step-by-step analysis not only provides the answer to the specific question but also reinforces the importance of understanding the fundamental properties of equality in mathematics. Mastering these properties is crucial for success in algebra and beyond. By carefully examining each step and considering the underlying principles, we can confidently navigate the world of equations and solve them with precision.
In the intricate world of algebra, solving equations is a fundamental skill. It requires a clear understanding of the properties that govern mathematical operations and transformations. This article delves into a specific equation-solving scenario, dissecting the steps taken and identifying the key property used in a crucial transformation. By examining the process in detail, we aim to solidify your understanding of algebraic principles and enhance your problem-solving abilities.
Analyzing Simon's Algebraic Steps
Let's revisit the equation-solving process presented, focusing on the transition between Step 3 and Step 4. Understanding the context of each step is crucial for identifying the property at play.
Step 1: -6(x + 7) = 2(x - 11)
This is the initial equation, the starting point of our algebraic journey. It involves parentheses, which suggests that the distributive property will likely be applied in the next step. This initial form sets the stage for simplification and the eventual isolation of the variable 'x.'
Step 2: -6x - 42 = 2x - 22
In this step, the distributive property is skillfully employed. The -6 is multiplied by both 'x' and '7' within the parentheses on the left side, resulting in -6x - 42. Similarly, the 2 is multiplied by both 'x' and '-11' on the right side, yielding 2x - 22. This expansion eliminates the parentheses and makes the equation easier to manipulate.
Step 3: -8x - 42 = -22
Here, the equation is further simplified by combining like terms. Specifically, 2x is subtracted from both sides of the equation. This move consolidates the 'x' terms on one side, bringing us closer to isolating the variable and finding its value. The resulting equation, -8x - 42 = -22, is a significant step towards the solution.
Step 4: -8x = 20
This is the focal point of our analysis. The task is to identify the property that allows us to transform the equation from Step 3 to Step 4. The key observation is the disappearance of the -42 term on the left side. This strongly suggests that a specific property of equality has been applied.
Unveiling the Property in Action The Addition Property of Equality
The transition from Step 3 to Step 4 hinges on a fundamental principle of algebra the addition property of equality. This property is a cornerstone of equation solving, allowing us to manipulate equations while maintaining their balance.
The Essence of the Addition Property of Equality
The addition property of equality states that if the same value is added to both sides of an equation, the equality is preserved. In mathematical terms, if A = B, then A + C = B + C. This seemingly simple concept is a powerful tool for isolating variables and solving equations.
Let's examine how this property applies to the transformation between Step 3 and Step 4:
Step 3: -8x - 42 = -22
Step 4: -8x = 20
The key change is the elimination of the -42 term on the left side. This can be achieved by adding 42 to both sides of the equation:
-8x - 42 + 42 = -22 + 42
This simplifies to:
-8x = 20
Thus, the addition property of equality is the property that justifies this transformation. By adding 42 to both sides, we successfully isolated the term containing 'x' and moved closer to the final solution.
Ruling Out Other Properties A Matter of Precision
While the addition property of equality is the correct answer, it's important to understand why other properties are not applicable in this scenario. This exercise reinforces our understanding of each property's specific use.
- Subtraction Property of Equality: This property involves subtracting the same value from both sides, which is the inverse operation of what occurred in this step.
- Multiplication Property of Equality: This property involves multiplying both sides by the same value. No multiplication was performed to get from Step 3 to Step 4.
- Division Property of Equality: This property involves dividing both sides by the same value. Again, no division was used in this step.
The addition property of equality is the only property that accurately reflects the transformation observed between Step 3 and Step 4.
Mastery Through Understanding The Addition Property's Significance
In conclusion, the property used to get from Step 3 to Step 4 in the equation-solving process is the addition property of equality. This property is a fundamental tool in algebra, allowing us to add the same value to both sides of an equation without disrupting its balance. It's a cornerstone of equation solving, and understanding its application is crucial for success in algebra.
By meticulously analyzing each step and considering the underlying principles, we gain a deeper appreciation for the elegance and precision of mathematics. This understanding not only answers the specific question but also strengthens our overall problem-solving skills. The addition property of equality, along with other properties, forms the foundation upon which more complex algebraic concepts are built. A solid grasp of these fundamentals is essential for navigating the world of equations with confidence.
