Fiona's Step-by-Step Solution To 1/2 - 1/3(6x - 3) = -13/2

Hey everyone! Let's dive into a mathematics problem today where we'll analyze how Fiona tackled an equation. We're going to break down each step of her solution, making sure we understand exactly what she did and why. This is a fantastic way to sharpen our equation-solving skills and boost our understanding of algebra. So, grab your thinking caps, and let's get started!

The Problem Unveiled

The original equation Fiona needed to solve was:

$\frac{1}{2}-\frac{1}{3}(6 x-3)=-\frac{13}{2}$

This equation involves fractions, parentheses, and variables, which might seem intimidating at first. But don't worry! We'll take it step by step, just like Fiona did, to make it super clear. Our goal is to understand the logic behind each move and how it brings us closer to finding the value of 'x'. Remember, practice makes perfect, and breaking down complex problems into smaller, manageable steps is a key strategy in mathematics.

Fiona's Solution Decoded

Fiona's solution is presented in a table format, which is a great way to organize the steps and keep track of the resulting equations. Let's go through each step in detail.

Step 1: Applying the Distributive Property

In the first step, Fiona uses the distributive property to simplify the equation. This property is crucial when dealing with expressions inside parentheses. The distributive property states that a(b + c) = ab + ac. In our case, we need to distribute the $-\frac{1}{3}$ across the terms inside the parentheses (6x - 3).

So, $-\frac{1}{3} * 6x = -2x$ and $-\frac{1}{3} * -3 = +1$. Applying this to the original equation, we get: $\frac{1}{2} - 2x + 1 = -\frac{13}{2}$.

Fiona's result for this step is $\frac{1}{2}-2 x+1=-\frac{13}{2}$. This looks correct so far! The distributive property was applied accurately, and the resulting equation is a simplified version of the original.

Step 2: Combining Like Terms

Combining like terms is a fundamental algebraic technique that simplifies equations by grouping similar elements together. In this context, like terms are constants (numbers without variables) or terms with the same variable raised to the same power. Fiona's equation after the distributive property is $\frac{1}{2} - 2x + 1 = -\frac{13}{2}$. We can identify two constant terms on the left side: $\frac{1}{2}$ and $+1$. To combine them, we simply add them together.

$\frac{1}{2} + 1$ can be rewritten as $\frac{1}{2} + \frac{2}{2}$, which equals $\frac{3}{2}$. So, the equation becomes $\frac{3}{2} - 2x = -\frac{13}{2}$. This step makes the equation cleaner and easier to work with. By combining like terms, we reduce the number of individual elements in the equation, making subsequent steps like isolating the variable more straightforward. This is a common and essential practice in solving algebraic equations, helping to streamline the process and minimize potential errors.

Step 3: Isolating the Variable Term

Isolating the variable term is a critical step in solving equations, as it brings us closer to finding the value of the unknown variable. In this case, we want to isolate the term containing 'x,' which is -2x. To do this, we need to eliminate any other terms on the same side of the equation. From the previous step, we have the equation $\frac{3}{2} - 2x = -\frac{13}{2}$. The term we need to eliminate is $\frac{3}{2}$.

To remove $\frac{3}{2}$, we perform the opposite operation, which is subtracting $\frac{3}{2}$ from both sides of the equation. This maintains the balance of the equation, ensuring that we're performing a valid algebraic manipulation. 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, $\frac{3}{2}$ and $-\frac{3}{2}$ cancel each other out, leaving us with -2x. On the right side, we have $-\frac{13}{2} - \frac{3}{2}$, which equals $-\frac{16}{2}$ or -8. So, the equation simplifies to -2x = -8. This step is crucial because it separates the variable term from the constants, making the final step of solving for 'x' much easier. Isolating the variable term is a fundamental technique in algebra and is used extensively in solving various types of equations.

Step 4: Solving for x

The final step in solving for 'x' involves isolating the variable completely. We've reached the equation -2x = -8. Now, we need to get 'x' by itself. To do this, we perform the opposite operation of what's being done to 'x'. Currently, 'x' is being multiplied by -2. So, to isolate 'x', we need to divide both sides of the equation by -2. This maintains the balance of the equation and ensures we're performing a valid algebraic operation.

Dividing both sides by -2 gives us: $\frac{-2x}{-2} = \frac{-8}{-2}$. On the left side, -2x divided by -2 simplifies to x. On the right side, -8 divided by -2 equals 4. Therefore, the solution is x = 4. This final step is the culmination of all the previous steps, where we systematically simplified the equation to reveal the value of the unknown variable. Solving for 'x' is the ultimate goal in many algebraic problems, and it allows us to find the specific value that satisfies the equation.

Conclusion: Fiona's Triumph

So, after carefully examining each step, we can see that Fiona has correctly solved the equation. Her solution, x = 4, is the accurate answer. By using the distributive property, combining like terms, and isolating the variable, she navigated the problem with precision. Great job, Fiona! And great job to you guys for following along. Keep practicing, and you'll become equation-solving pros in no time!