Kate's Equation Solving Journey A Mathematical Exploration

Kate embarked on a mathematical journey to solve the equation $\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)$. Her initial steps involved adding 2 to both sides, which led her to the simplified equation $4x = 3x$. This article delves into Kate's problem-solving process, explores the implications of her steps, and discusses the best solution to illustrate the equation's behavior. We will dissect the equation, analyze the transformations, and ultimately, arrive at the solution that best elucidates the mathematical principles at play. This exploration will not only showcase Kate's work but also provide a comprehensive understanding of the underlying algebraic concepts. The goal is to provide a detailed explanation that is both mathematically sound and accessible to a broad audience, making the learning experience engaging and effective. We will begin by examining the original equation and then meticulously trace Kate's steps to fully grasp the solution's path. This meticulous approach will allow us to identify the key transformations and understand their impact on the equation's solution.

H2: Unveiling the Initial Equation

Let's start by examining the original equation: $\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)$. This equation represents a fundamental algebraic problem where we aim to find the value(s) of 'x' that satisfy the equality. To truly understand the solution, we need to break down the equation and analyze each component. The left-hand side of the equation involves multiplying a fraction, $\frac{2}{3}$, with an expression in parentheses, $(6x - 3)$. Similarly, the right-hand side involves multiplying a fraction, $\frac{1}{2}$, with another expression in parentheses, $(6x - 4)$. The presence of fractions and parentheses indicates that the initial steps in solving this equation will likely involve distribution and simplification. Before we proceed, it's crucial to recognize that the equation is a statement of balance. The left-hand side and the right-hand side must be equal for the equation to hold true. Our task is to manipulate the equation while maintaining this balance, ultimately isolating 'x' to find its value. The coefficients and constants within the equation play a crucial role in determining the solution. The fractions $\frac{2}{3}$ and $\frac{1}{2}$ will influence the scaling of the expressions, while the constants -3 and -4 will affect the position of the solution on the number line. Therefore, a thorough understanding of these elements is essential for navigating the solution process effectively. We will now delve into the specific steps Kate undertook, starting with the addition of 2 to both sides, to trace her path towards the simplified equation.

H2: Kate's Transformation Adding 2 to Both Sides

Kate's initial move was to add 2 to both sides of the equation. While this might seem like an arbitrary step at first glance, it's crucial to understand the underlying algebraic principle. Adding the same value to both sides of an equation maintains the equality. This is a fundamental rule in algebra, ensuring that the solution remains unchanged throughout the transformation. To analyze this step, let's write out the equation before and after the addition: Original equation: $\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)$ After adding 2 to both sides: $\frac{2}{3}(6x - 3) + 2 = \frac{1}{2}(6x - 4) + 2$ Now, we need to carefully examine how this addition leads to the simplified equation $4x = 3x$. This requires us to perform the arithmetic operations and simplify both sides of the equation. The left-hand side involves adding 2 to the expression $\frac{2}{3}(6x - 3)$. To do this, we need to distribute the $\frac{2}{3}$ and then combine like terms. Similarly, on the right-hand side, we need to distribute the $\frac{1}{2}$ and then add 2. The process of simplifying these expressions will reveal the intermediate steps that connect the initial equation to the resulting equation. It's important to note that this step may not be the most direct way to solve the equation. However, it provides a valuable opportunity to illustrate how algebraic manipulations can transform an equation while preserving its solution. The resulting equation, $4x = 3x$, presents a new perspective on the problem, highlighting the relationship between 'x' on both sides. We will now analyze the implications of this equation and explore how it ultimately leads to the solution.

H2: The Resulting Equation 4x = 3x and Its Implications

The equation $4x = 3x$ is a simplified form of the original equation, resulting from Kate's addition of 2 to both sides and subsequent simplification. This equation is particularly interesting because it presents a direct comparison between multiples of 'x'. To find the solution, we need to isolate 'x'. A common approach would be to subtract $3x$ from both sides of the equation. This gives us: $4x - 3x = 3x - 3x$ which simplifies to $x = 0$ This reveals that the solution to the equation is $x = 0$. However, it's crucial to understand why this is the case and what it means in the context of the original equation. The equation $4x = 3x$ essentially states that four times a certain number is equal to three times the same number. This is only possible if the number is zero. Any other value for 'x' would violate this equality. Now, let's consider the implications for the original equation. If $x = 0$, then we can substitute this value back into the original equation to verify the solution. Substituting $x = 0$ into $\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)$, we get: $\frac{2}{3}(6(0) - 3) = \frac{1}{2}(6(0) - 4)$ $\frac{2}{3}(-3) = \frac{1}{2}(-4)$ $-2 = -2$ This confirms that $x = 0$ is indeed the solution to the original equation. The equation $4x = 3x$ provides a concise representation of this solution, highlighting the specific value of 'x' that satisfies the equation. This step in Kate's journey underscores the power of algebraic manipulation in transforming an equation into a more manageable form, ultimately leading to the solution.

H2: Illustrating the Solution Graphically and Algebraically

To best illustrate what happens to the solution when solving the equation, we can employ both graphical and algebraic methods. This dual approach provides a comprehensive understanding of the mathematical principles involved. Graphically, we can represent each side of the original equation as a separate function: $y_1 = \frac{2}{3}(6x - 3)$ $y_2 = \frac{1}{2}(6x - 4)$ These are both linear functions, and their graphs are straight lines. The solution to the equation is the x-coordinate of the point where the two lines intersect. By plotting these lines on a coordinate plane, we can visually identify the intersection point and confirm that it occurs at $x = 0$. This graphical representation provides a clear and intuitive understanding of the solution. Algebraically, we can trace the steps involved in solving the equation, highlighting how each transformation affects the solution. Starting with the original equation: $\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)$ We can first distribute the fractions: $4x - 2 = 3x - 2$ Then, we can add 2 to both sides, as Kate did: $4x = 3x$ Finally, we can subtract $3x$ from both sides: $x = 0$ This algebraic process demonstrates how each step preserves the solution, leading us to the final answer. The step of adding 2 to both sides, while not the most direct path to the solution, is mathematically valid and can be shown to preserve the solution. The graphical method provides a visual confirmation of the solution, while the algebraic method provides a step-by-step logical progression. Together, these methods offer a robust illustration of the solution process and the underlying mathematical principles. In conclusion, the solution $x = 0$ is best illustrated by a combination of graphical representation and a step-by-step algebraic derivation, showcasing the consistency of the mathematical transformations.

H2: Conclusion Kate's Equation Solving Success

In conclusion, Kate's journey to solve the equation $\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)$ provides a valuable illustration of algebraic problem-solving. Her initial step of adding 2 to both sides, while not the most conventional, ultimately led to the simplified equation $4x = 3x$, which clearly reveals the solution $x = 0$. This article has explored Kate's process in detail, analyzing the implications of each step and highlighting the underlying mathematical principles. We have demonstrated how the solution can be verified both algebraically and graphically, providing a comprehensive understanding of the equation's behavior. The graphical representation offers a visual confirmation of the solution, while the algebraic steps provide a logical and rigorous derivation. The combination of these methods underscores the consistency and validity of the mathematical transformations involved. Kate's work exemplifies the power of algebraic manipulation in simplifying equations and revealing their solution. The process of adding 2 to both sides, although not the most direct approach, showcases the flexibility of algebraic techniques and the importance of maintaining equality throughout the transformations. The resulting equation, $4x = 3x$, serves as a concise representation of the solution, emphasizing the specific value of 'x' that satisfies the equation. By understanding Kate's journey and the mathematical concepts involved, we gain a deeper appreciation for the art of equation solving and the elegance of algebraic reasoning. This exploration not only provides a solution to the specific equation but also enhances our overall understanding of mathematical problem-solving strategies.