Rewriting Equations: Deconstructing $\frac{4x}{3x-12} + 3 = 10$

Unlocking the secrets of algebraic equations often involves breaking them down into simpler, more manageable parts. When faced with a complex equation like $\frac{4x}{3x-12} + 3 = 10$, rewriting it into multiple equations joined by the logical connective "and" can illuminate the underlying structure and potential solutions. This approach isn't just about manipulating symbols; it's about gaining a deeper understanding of the relationships between variables and constants. In this article, we'll delve into the process of deconstructing this equation, exploring the mathematical principles at play and providing a clear, step-by-step guide to achieving the desired transformation. Our focus will be on ensuring each step is logically sound and easy to follow, making this process accessible to anyone eager to enhance their algebraic skills.

Understanding the Equation's Structure

Before we dive into the manipulation, let's take a moment to understand the structure of the equation $\frac{4x}{3x-12} + 3 = 10$. At its core, this equation involves a rational expression, $\frac{4x}{3x-12}$, a constant term, 3, and a target value, 10. The variable we're trying to solve for is 'x'. To effectively rewrite this equation, we need to consider the restrictions imposed by the rational expression and how they impact the possible solutions. Specifically, the denominator, $3x - 12$, cannot be equal to zero, as division by zero is undefined in mathematics. This leads to our first crucial equation, which we'll combine with the simplified form of the original equation. Recognizing these constraints is paramount because it ensures we don't arrive at solutions that are mathematically invalid. Moreover, understanding the domain of the variable 'x' is essential for the overall integrity of our solution process. We're not just looking for any number that satisfies the equation; we're looking for a number that satisfies it within the boundaries of mathematical rules. Therefore, our initial step in rewriting the equation involves explicitly stating this restriction, setting the stage for a more accurate and comprehensive solution. This careful attention to detail is what distinguishes a superficial understanding of algebra from a deep, working knowledge.

Step-by-Step Rewriting Process

Now, let's embark on the step-by-step journey of rewriting the equation $\frac{4x}{3x-12} + 3 = 10$ into two equations joined by "and".

1. Identify the Restriction

As we discussed earlier, the denominator of the rational expression cannot be zero. Therefore, we begin by setting the denominator not equal to zero:

$3x - 12 \neq 0$

This inequality forms our first equation, which we'll ultimately combine with the simplified form of the original equation using the "and" connective. This step is absolutely critical because it defines the valid domain for 'x'. Ignoring this restriction could lead to extraneous solutions, which are values that satisfy the transformed equation but not the original one. By explicitly stating this condition, we ensure that any solution we find will be mathematically sound. It's a bit like setting the rules of the game before we start playing, ensuring fairness and accuracy in our results. Furthermore, this initial step highlights the importance of considering the context of mathematical expressions, particularly rational expressions, where denominators play a crucial role in determining the validity of solutions. This approach of identifying restrictions upfront is a cornerstone of robust algebraic problem-solving.

2. Simplify the Original Equation

Next, we'll simplify the original equation to isolate the rational expression and make it easier to work with. Subtract 3 from both sides of the equation:

$\frac{4x}{3x-12} + 3 - 3 = 10 - 3$

This simplifies to:

$\frac{4x}{3x-12} = 7$

This simplified equation represents the core relationship between 'x' and the constants in the original equation, excluding the domain restriction. This step is crucial for making the equation more approachable and less cluttered. By isolating the rational expression, we can focus on solving for 'x' in a more direct manner. It's like clearing away the underbrush in a forest to see the trees more clearly. This simplification not only makes the equation easier to solve but also reveals the underlying structure more effectively. We've essentially stripped away the extra layer, allowing us to focus on the heart of the problem. This ability to simplify complex expressions is a hallmark of algebraic proficiency, enabling us to tackle more challenging problems with greater confidence. Moreover, this simplified form sets the stage for the next steps in our rewriting process, bringing us closer to our goal of expressing the original equation as two equations joined by "and".

3. Eliminate the Fraction

To further simplify, we'll eliminate the fraction by multiplying both sides of the equation by the denominator, $3x - 12$:

$\frac{4x}{3x-12} * (3x - 12) = 7 * (3x - 12)$

This results in:

$4x = 7(3x - 12)$

This step is a pivotal moment in our equation-rewriting journey. By eliminating the fraction, we transform a potentially daunting rational equation into a more familiar linear equation. This is a common and powerful technique in algebra, allowing us to work with whole numbers and variables, rather than fractions. It's akin to converting a complex map into a simpler, more manageable one, making our route to the solution clearer. This transformation is not just about making the equation look simpler; it's about changing its fundamental form while preserving its mathematical integrity. We're essentially performing an operation that maintains the balance of the equation, ensuring that the solutions remain the same. This ability to manipulate equations while preserving their solutions is a key skill in algebra, opening doors to solving a wider range of problems. Furthermore, this step prepares us for the final algebraic manipulations needed to isolate 'x' and find its value, bringing us closer to a complete solution.

4. Distribute and Rearrange

Now, let's distribute the 7 on the right side of the equation:

$4x = 21x - 84$

Next, we'll rearrange the equation to group the 'x' terms on one side and the constants on the other. Subtract $21x$ from both sides:

$4x - 21x = -84$

This simplifies to:

$-17x = -84$

This step is where we begin to see the fruits of our earlier labors. By distributing and rearranging, we're isolating the variable 'x' and bringing the equation closer to its solved form. It's like assembling the pieces of a puzzle, carefully positioning each element to reveal the final picture. This process of rearranging terms is fundamental to solving algebraic equations, allowing us to separate the known from the unknown. Each operation we perform, whether it's distribution or subtraction, is a deliberate step towards simplifying the equation and revealing the value of 'x'. This meticulous approach is what ensures accuracy and clarity in our solution. Furthermore, this rearrangement sets the stage for the final step in solving for 'x', where we'll perform one last operation to isolate the variable completely. This methodical progression is a testament to the power of algebraic manipulation in untangling complex relationships and arriving at solutions.

5. Solve for x

Finally, we'll solve for 'x' by dividing both sides of the equation by -17:

$\frac{-17x}{-17} = \frac{-84}{-17}$

This gives us:

$x = \frac{84}{17}$

This step marks the culmination of our algebraic journey. After a series of meticulous transformations, we've successfully isolated 'x' and found its value. It's like reaching the summit of a mountain after a challenging climb, a moment of triumph and clarity. This final division is the key that unlocks the solution, revealing the numerical value that satisfies the equation. However, our work isn't quite done yet. We need to remember the initial restriction we identified, the condition that $3x - 12 \neq 0$. We must check whether our solution, $x = \frac{84}{17}$, satisfies this condition. This verification step is essential to ensure that our solution is not extraneous and is mathematically valid. It's a reminder that solving equations is not just about manipulating symbols; it's about understanding the underlying mathematical principles and constraints. This complete process, from simplification to verification, exemplifies the rigor and precision of algebraic problem-solving.

6. Express as Two Equations Joined by "and"

Now, we can express the original equation as two equations joined by "and":

1. $3x - 12 \neq 0$
2. $x = \frac{84}{17}$

This final step encapsulates the essence of our rewriting process. We've successfully deconstructed the original equation into two distinct statements, connected by the logical connective "and". This representation highlights both the solution to the equation and the condition under which that solution is valid. It's like presenting a complete picture, with both the foreground and the background in clear focus. The first equation, $3x - 12 \neq 0$, defines the domain of 'x', the set of values for which the original equation is meaningful. The second equation, $x = \frac{84}{17}$, provides the specific value of 'x' that satisfies the equation within that domain. This two-part expression provides a more nuanced understanding of the solution, emphasizing the importance of context and constraints in mathematical problem-solving. Furthermore, this representation demonstrates the power of logical connectives in expressing complex mathematical relationships, showcasing the elegance and precision of mathematical language. This culmination of our efforts highlights the value of breaking down complex problems into simpler components, a strategy that is applicable not only in mathematics but also in many other areas of life.

Verification

It's crucial to verify that our solution, $x = \frac{84}{17}$, satisfies the restriction $3x - 12 \neq 0$:

$3(\frac{84}{17}) - 12 = \frac{252}{17} - \frac{204}{17} = \frac{48}{17} \neq 0$

Since the restriction is satisfied, our solution is valid. This verification process is a critical safety net in mathematical problem-solving. It's the final check that ensures our answer is not only mathematically correct but also logically sound within the context of the original equation. By substituting our solution back into the restricted expression, we confirm that it doesn't violate any of the fundamental rules of mathematics, such as division by zero. This step is not merely a formality; it's an integral part of the solution, guarding against extraneous solutions that may arise from algebraic manipulations. It's like testing the strength of a bridge before allowing traffic to cross, ensuring that it can bear the load. This commitment to verification demonstrates a deep understanding of mathematical principles and a dedication to accuracy. Furthermore, this process reinforces the importance of attention to detail and the interconnectedness of different parts of a mathematical problem, highlighting the holistic nature of mathematical thinking.

Conclusion

In conclusion, we've successfully rewritten the equation $\frac{4x}{3x-12} + 3 = 10$ as two equations joined by "and":

1. $3x - 12 \neq 0$
2. $x = \frac{84}{17}$

This process highlights the importance of understanding the structure of equations and the constraints they impose. By carefully considering the domain of the variable and applying algebraic manipulations, we can effectively rewrite complex equations into simpler, more manageable forms. This journey through the equation-rewriting process has been a testament to the power of algebraic thinking. We've not only found the solution to a specific problem but also reinforced fundamental mathematical principles and techniques. The ability to deconstruct complex equations into simpler components is a valuable skill, applicable to a wide range of mathematical challenges. Moreover, this process underscores the importance of precision, attention to detail, and the verification of solutions. As we've seen, mathematics is not just about finding the right answer; it's about understanding the reasoning behind it and ensuring its validity within the broader mathematical framework. This holistic approach to problem-solving is what truly empowers us to tackle complex challenges with confidence and clarity. To further enhance your understanding of algebraic equations and their solutions, consider exploring resources like Khan Academy's Algebra I course.