Justification For Step 2 In Equation Solving A Detailed Explanation
Let's dive deep into the fascinating world of algebraic equations and explore the logical steps involved in solving them. In mathematics, each step we take needs to be justified by a fundamental principle or property. This ensures that our solutions are not only correct but also logically sound. Today, we'll dissect a specific problem and unravel the justification behind one of its crucial steps.
Understanding the Problem
We are presented with the following equation:
10x - 25 - 3x = 4x - 1
The solution process begins with:
Step 1: 7x - 25 = 4x - 1
Step 2: 7x = 4x + 24
The core question we aim to answer is: What is the justification for Step 2 in the solution process?
To answer this, we will scrutinize the transition from Step 1 to Step 2, identify the mathematical operation performed, and then connect it to the relevant property of equality. The options provided are:
- A. The multiplication property of equality
- B. The division property of equality
To accurately identify the correct answer, we'll need to delve into the properties of equality and understand how they apply in manipulating equations.
Deciphering Step 1 The Foundation of Our Solution
Before we can justify Step 2, it's crucial to understand how we arrived at Step 1. The given equation is:
10x - 25 - 3x = 4x - 1
Step 1 simplifies this equation to:
7x - 25 = 4x - 1
This transition involves combining like terms on the left side of the equation. Specifically, the terms 10x and -3x are combined. This is a direct application of the distributive property and the commutative property of addition. The distributive property allows us to combine coefficients of the same variable, and the commutative property allows us to rearrange terms within the expression.
- Distributive Property: a(b + c) = ab + ac
- Commutative Property of Addition: a + b = b + a
In our case, we effectively have (10 - 3)x, which simplifies to 7x. Thus, the left side of the equation becomes 7x - 25. The right side, 4x - 1, remains unchanged. This initial simplification is a fundamental step in solving algebraic equations, making the equation more manageable for subsequent operations. Understanding this groundwork is essential for appreciating the justification behind Step 2.
Unveiling Step 2 The Critical Transition
Now, let's focus on the transition from Step 1 to Step 2. We have:
Step 1: 7x - 25 = 4x - 1
Step 2: 7x = 4x + 24
Observing this transition closely, we notice that the -25 term on the left side of the equation in Step 1 has disappeared in Step 2. Instead, we see a +24 on the right side. This strongly suggests that a specific operation was performed on both sides of the equation to achieve this transformation.
The key here is to recognize that we've effectively eliminated the constant term on the left side by adding 25 to both sides of the equation. This is a critical concept in solving equations – maintaining balance by performing the same operation on both sides.
- Starting with
7x - 25 = 4x - 1 - We add
25to both sides:7x - 25 + 25 = 4x - 1 + 25 - This simplifies to
7x = 4x + 24
This addition is justified by the addition property of equality, which is a cornerstone principle in algebra. Let's explore this property in detail.
The Addition Property of Equality The Key to Unlocking Step 2
The addition property of equality is a fundamental principle in algebra that dictates how we can manipulate equations without altering their fundamental truth. This property states:
If a = b, then a + c = b + c for any real number c.
In simpler terms, this means that we can add the same value to both sides of an equation, and the equation will remain balanced and true. This property is crucial because it allows us to isolate variables and simplify equations systematically.
In the context of our problem, the addition property of equality is precisely what justifies Step 2. We added 25 to both sides of the equation in Step 1 (7x - 25 = 4x - 1) to eliminate the -25 term on the left side. This maintains the equality and moves us closer to isolating the variable x. The application of this property is not just a mathematical trick; it's a logical step grounded in a core algebraic principle.
Understanding the addition property of equality is paramount for anyone learning algebra. It provides the foundation for solving a wide range of equations and is a crucial tool in mathematical problem-solving.
Debunking the Distractors Why Other Options Don't Fit
To solidify our understanding, let's briefly examine why the other options provided are not the correct justification for Step 2:
- A. The multiplication property of equality: This property states that if a = b, then ac = bc for any real number c. While this property is essential for solving equations, it involves multiplying both sides by the same value. In our transition from Step 1 to Step 2, we added a value, not multiplied, making this option incorrect.
- B. The division property of equality: This property states that if a = b, then a/c = b/c for any non-zero real number c. Similar to the multiplication property, this involves dividing both sides by the same value. Since we added a value in Step 2, this option is also incorrect.
By carefully analyzing the operation performed and comparing it to the properties of equality, we can confidently rule out these options and reinforce our understanding of why the addition property of equality is the correct justification.
Conclusion Solidifying Our Understanding
In conclusion, the justification for Step 2 in the solution process is the implicit use of the addition property of equality. By adding 25 to both sides of the equation, we maintained balance and progressed towards isolating the variable. This step is not arbitrary; it's a direct application of a fundamental algebraic principle.
Understanding the properties of equality is crucial for mastering algebra. These properties provide the logical foundation for manipulating equations and solving for unknowns. By dissecting each step and justifying it with a specific property, we develop a deeper and more robust understanding of mathematics.
So, the correct answer is implicitly the addition property of equality and while it wasn't explicitly listed as an option, understanding this principle is key to solving the problem. This exercise highlights the importance of not just finding the answer, but also understanding the 'why' behind each step in the solution process.
This detailed exploration should provide a clear understanding of the justification behind Step 2 and reinforce the importance of the addition property of equality in solving algebraic equations. Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying logic and principles.
What property justifies the transition from step 1 to step 2 in the equation 10x - 25 - 3x = 4x - 1, where step 1 is 7x - 25 = 4x - 1 and step 2 is 7x = 4x + 24?
Justification for Step 2 in Equation Solving A Detailed Explanation
