Justification For Step 4 In Solving The Equation (9/2)b + 11 - (5/6)b = B + 2

In this article, we will delve into the process of solving an algebraic equation and focus specifically on understanding the justification for a particular step. We'll dissect the given equation, trace the solution pathway, and clarify the logic behind each transformation. Our goal is to provide a comprehensive explanation that not only answers the question but also enhances your overall understanding of algebraic problem-solving.

The Initial Equation: A Starting Point

Let's begin by examining the initial equation that sets the stage for our problem:

$\frac{9}{2} b+11-\frac{5}{6} b=b+2$

This equation involves a variable, b, along with several constants and fractions. Our objective is to isolate b on one side of the equation to determine its value. To achieve this, we will systematically simplify the equation by combining like terms and performing valid algebraic operations. The beauty of algebra lies in its structured approach, where each step is justified by fundamental mathematical principles.

Tracing the Steps Towards the Solution

The provided solution excerpt skips steps 2 and 3, jumping directly to step 4. This leap can make it challenging to grasp the underlying logic. Therefore, we will reconstruct the missing steps to provide a complete and clear solution path. This reconstruction will allow us to meticulously analyze how the equation is transformed from one form to another, ultimately leading us to the critical step 4 and its justification.

Reconstructing the Missing Steps: A Detailed Walkthrough

To effectively explain the justification for step 4, we must first fill in the missing steps 2 and 3. These steps likely involve combining the terms containing the variable b on the left side of the equation. Let's embark on this reconstruction process:

Step 1: Original Equation

As given, our starting point is:

$\frac{9}{2} b+11-\frac{5}{6} b=b+2$

Step 2: Combining 'b' Terms

The next logical step is to combine the terms involving b on the left side of the equation. These terms are $\frac{9}{2} b$ and $-\frac{5}{6} b$. To combine them, we need a common denominator. The least common multiple of 2 and 6 is 6. Thus, we rewrite the fractions:

$\frac{9}{2} b = \frac{9 * 3}{2 * 3} b = \frac{27}{6} b$

Now we can rewrite the equation, replacing $\frac{9}{2} b$ with $\frac{27}{6} b$:

$\frac{27}{6} b + 11 - \frac{5}{6} b = b + 2$

Combining the b terms, we get:

$\frac{27}{6} b - \frac{5}{6} b = \frac{27 - 5}{6} b = \frac{22}{6} b$

So, step 2 can be represented as:

$\frac{22}{6} b + 11 = b + 2$

Step 3: Preparing for Isolation

Step 3 could involve either moving the constant term to the right side of the equation or moving the b term on the right side to the left. Let's subtract b from both sides to bring the b terms together on the left:

$\frac{22}{6} b + 11 - b = b + 2 - b$

This simplifies to:

$\frac{22}{6} b - b + 11 = 2$

To combine the b terms, we need to express b as a fraction with a denominator of 6:

$b = \frac{6}{6} b$

Therefore, we have:

$\frac{22}{6} b - \frac{6}{6} b + 11 = 2$

Which simplifies to:

$\frac{16}{6} b + 11 = 2$

Alternatively, Step 3 could involve subtracting 11 from both sides:

$\frac{22}{6} b + 11 - 11 = b + 2 - 11$

Which simplifies to:

$\frac{22}{6} b = b - 9$

It seems more logical that step 3 is $\frac{22}{6} b + 11 = b + 2$, as this is what was derived in our reconstruction of step 2.

Step 4: The Pivotal Transformation

Now, let's examine the given step 4:

$\frac{22}{6} b+11=b+2$

This step is the result of combining the b terms on the left side of the original equation, as we derived in our reconstruction of Step 2.

Justification for Step 4: Unveiling the Rationale

The justification for step 4 lies in the fundamental principles of algebraic manipulation. Specifically, it is based on the properties of equality and the combination of like terms. Let's break this down:

• Properties of Equality: These properties allow us to perform the same operation on both sides of an equation without changing its solution. In this case, we've implicitly used the additive property of equality (adding or subtracting the same quantity from both sides) and the distributive property (to combine coefficients). However, in this particular step, no operation was performed on both sides of the equation. Instead, it involves simplification on one side.
• Combining Like Terms: This principle states that terms with the same variable raised to the same power can be combined by adding or subtracting their coefficients. In our equation, $\frac{9}{2} b$ and $-\frac{5}{6} b$ are like terms because they both contain the variable b raised to the power of 1. To combine them, we found a common denominator and added their coefficients, as demonstrated in the reconstruction of Step 2.

Therefore, the explicit justification for step 4 is the combination of like terms on the left-hand side of the equation. We transformed $\frac{9}{2} b+11-\frac{5}{6} b$ into $\frac{22}{6} b+11$ by finding a common denominator for the fractions, adding the coefficients of the b terms, and maintaining the constant term.

Why This Step Matters: Building Towards the Solution

Step 4 is crucial because it simplifies the equation, bringing us closer to isolating the variable b. By combining the b terms, we reduce the number of terms in the equation, making it easier to manipulate. This simplification is a standard strategy in solving algebraic equations and lays the groundwork for subsequent steps that will ultimately lead to the solution for b.

Continuing the Solution: A Glimpse Ahead

From step 4, we would typically proceed by isolating the b term further. This might involve subtracting the constant term (11) from both sides of the equation and then dividing both sides by the coefficient of b ($\frac{22}{6}$). These steps would follow the same principles of equality, ensuring that the solution remains valid throughout the process.

Conclusion: The Power of Justification in Algebra

Understanding the justification for each step in solving an algebraic equation is paramount. It's not just about arriving at the correct answer; it's about comprehending the underlying mathematical principles that govern the process. In the case of step 4, the justification lies in the combination of like terms, a fundamental concept in algebra. By grasping these principles, we can approach algebraic problems with confidence and clarity.

By reconstructing the missing steps and meticulously analyzing step 4, we've gained a deeper appreciation for the logic and structure of algebraic problem-solving. This understanding empowers us to tackle more complex equations and appreciate the elegance of mathematical reasoning.

This detailed explanation should provide a clear and comprehensive answer to the question, enhancing your understanding of algebraic equations and their solutions.

Keywords: algebraic equation, combining like terms, properties of equality, solving equations, justification for step 4