Solving Equations A Detailed Look At Fiona's Method

Fiona embarked on a mathematical journey to solve the equation $\frac{1}{2} - \frac{1}{3}(6x - 3) = -\frac{13}{2}$. Her solution process, meticulously documented, provides a fascinating glimpse into the application of algebraic principles. Let's delve into each step of Fiona's solution, analyze her methodology, and ensure the accuracy of her calculations. This detailed examination will not only validate Fiona's work but also serve as an instructive guide for anyone navigating the world of algebraic equations. We will break down each step, providing explanations and highlighting key concepts, making it easier to understand the underlying mathematical logic. This is crucial for students learning algebra and anyone who wants to refresh their understanding of equation-solving techniques.

Step 1: Unleashing the Distributive Property

The distributive property is a cornerstone of algebra, allowing us to simplify expressions by multiplying a term across a sum or difference within parentheses. Fiona initiated her solution by applying this fundamental property. In the original equation, $\frac{1}{2} - \frac{1}{3}(6x - 3) = -\frac{13}{2}$, the term $-\frac{1}{3}$ needs to be distributed across the binomial $(6x - 3)$. This means multiplying $-\frac{1}{3}$ by both $6x$ and $-3$.

The multiplication of $-\frac{1}{3}$ and $6x$ yields $-2x$. Next, multiplying $-\frac{1}{3}$ by $-3$ results in $+1$. Consequently, applying the distributive property transforms the equation into $\frac{1}{2} - 2x + 1 = -\frac{13}{2}$. This step is critical as it removes the parentheses, making the equation easier to manipulate and solve. Without correctly applying the distributive property, the subsequent steps would be based on a flawed foundation, leading to an incorrect solution. Understanding this step thoroughly is vital for mastering algebraic manipulations. It demonstrates the power of the distributive property in simplifying complex expressions and paving the way for a clear path to the solution. This initial simplification is often the key to unlocking the rest of the problem, allowing for easier combination of like terms and isolation of the variable. Fiona's correct application of this property sets the stage for a successful solution.

Step 2: Combining Like Terms - Simplifying the Equation

After effectively applying the distributive property, the next logical step in solving the equation is to combine like terms. This process involves identifying terms with the same variable or constant components and then adding or subtracting their coefficients. In Fiona's equation, $\frac{1}{2} - 2x + 1 = -\frac{13}{2}$, we can identify two constant terms: $\frac{1}{2}$ and $1$. To combine these, we need to find a common denominator, which in this case is 2. Converting 1 to a fraction with a denominator of 2 gives us $\frac{2}{2}$. Now, we can add the fractions: $\frac{1}{2} + \frac{2}{2} = \frac{3}{2}$.

Therefore, the equation simplifies to $\frac{3}{2} - 2x = -\frac{13}{2}$. Combining like terms is a crucial step in simplifying equations, as it reduces the number of terms and makes the equation easier to solve. It allows us to consolidate the constant terms on one side of the equation, paving the way for isolating the variable term. By combining the constant terms, Fiona has streamlined the equation, making it more manageable and setting the stage for the next step in the solution process. This meticulous attention to detail in simplifying the equation showcases a strong understanding of algebraic principles. The act of combining like terms is not just about numerical manipulation; it's about strategically organizing the equation to reveal the underlying structure and make the solution more apparent. This step highlights the importance of precision and careful calculation in algebraic problem-solving.

Step 3: Isolating the Variable Term - Moving Towards the Unknown

Having simplified the equation to $\frac{3}{2} - 2x = -\frac{13}{2}$, the focus now shifts to isolating the variable term, $-2x$. This involves strategically manipulating the equation to get the term containing the variable alone on one side. To achieve this, Fiona needs to eliminate the constant term, $\frac{3}{2}$, from the left side of the equation. The standard method for doing this is to subtract $\frac{3}{2}$ from both sides of the equation. This maintains the balance of the equation, ensuring that any operation performed on one side is also performed on the other, preserving the equality. Subtracting $\frac{3}{2}$ from both sides gives us: $\frac{3}{2} - 2x - \frac{3}{2} = -\frac{13}{2} - \frac{3}{2}$.

On the left side, the $\frac{3}{2}$ and $-\frac{3}{2}$ terms cancel each other out, leaving $-2x$. On the right side, we need to subtract the fractions. Since they already have a common denominator, we can simply subtract the numerators: $-\frac{13}{2} - \frac{3}{2} = -\frac{16}{2}$. Simplifying the fraction $-\frac{16}{2}$ gives us $-8$. Therefore, the equation now becomes $-2x = -8$. Isolating the variable term is a pivotal step in solving equations, as it brings us closer to determining the value of the unknown. By removing the constant term from the left side, Fiona has successfully created a direct relationship between the variable term and a constant value. This step underscores the importance of maintaining balance in equations and using inverse operations to isolate variables. The resulting equation, $-2x = -8$, is significantly simpler than the original and makes the final step of solving for $x$ much more straightforward.

Step 4: Solving for x - Unveiling the Solution

With the variable term isolated, the final step is to solve for $x$. Fiona's equation now stands as $-2x = -8$. To isolate $x$, we need to undo the multiplication by $-2$. The inverse operation of multiplication is division, so we divide both sides of the equation by $-2$. This maintains the equality of the equation and allows us to isolate $x$. Dividing both sides by $-2$ gives us: $\frac{-2x}{-2} = \frac{-8}{-2}$.

On the left side, the $-2$ in the numerator and denominator cancel each other out, leaving just $x$. On the right side, we have $\frac{-8}{-2}$, which simplifies to $4$. Therefore, the solution to the equation is $x = 4$. Solving for the variable is the culmination of all the previous steps. It's the moment where the value of the unknown is revealed, providing the answer to the problem. Fiona's successful isolation of $x$ and subsequent division demonstrate a clear understanding of algebraic principles and the ability to apply them accurately. This final step underscores the importance of inverse operations in equation-solving and the systematic approach required to arrive at the correct solution. The answer, $x = 4$, is the value that satisfies the original equation, making the equation a true statement. This final validation is a testament to the correctness of Fiona's solution process.

Conclusion: Fiona's Triumph in Equation Solving

In conclusion, Fiona's journey to solve the equation $\frac{1}{2} - \frac{1}{3}(6x - 3) = -\frac{13}{2}$ showcases a strong command of algebraic principles. Her meticulous application of the distributive property, skillful combination of like terms, strategic isolation of the variable term, and accurate solution for $x$ demonstrate a comprehensive understanding of equation-solving techniques. Each step in Fiona's solution process was executed with precision and clarity, leading to the correct answer, $x = 4$. This detailed analysis not only validates Fiona's solution but also serves as an excellent guide for anyone learning or reviewing algebraic concepts. The importance of each step, from the initial simplification to the final solution, has been highlighted, emphasizing the logical progression required to solve equations effectively. Fiona's success is a testament to the power of systematic problem-solving and the importance of mastering fundamental algebraic principles. Her journey serves as an inspiration and a valuable learning resource for students and math enthusiasts alike.