Solve For X: A Step-by-Step Equation Guide

Understanding the Equation

Welcome to our exploration of algebraic equations! Today, we're diving into a specific problem: finding the value of x in the equation $\frac{2}{2}\left(\frac{1}{2} x+12\right)-\frac{1}{2}\left(\frac{1}{3} x+14\right)-3$. This might look a bit intimidating with all the fractions and parentheses, but don't worry! We'll break it down piece by piece, making it as clear as day. Our main goal is to isolate x on one side of the equation. This means we need to use a series of logical steps, applying the rules of algebra, to simplify the expression and uncover the mystery value of x. Think of it like solving a puzzle where each step brings us closer to the final solution.

We'll start by simplifying the equation. This involves dealing with the coefficients outside the parentheses and then combining like terms. The order of operations (PEMDAS/BODMAS) is our best friend here. We'll tackle the parentheses first, then multiplication, and finally, addition and subtraction. Remember, every step must be performed correctly to ensure our final answer is accurate. It’s a journey of simplification, moving from a complex expression to a simple statement that reveals x's true value. We will cover the importance of each operation and how it affects the overall equation. By the end, you'll not only know the value of x but also understand why it's that value, giving you confidence in tackling similar problems in the future. Let's get started on this mathematical adventure!

Step 1: Simplify the Equation

The first crucial step in solving for x in the given equation is to simplify the expression by distributing the coefficients to the terms inside the parentheses. Our equation is: $\frac{2}{2}\left(\frac{1}{2} x+12\right)-\frac{1}{2}\left(\frac{1}{3} x+14\right)-3$. Let's tackle the first part: $\frac{2}{2}\left(\frac{1}{2} x+12\right)$. Since $\frac{2}{2}$ equals 1, this part simplifies to just $\left(\frac{1}{2} x+12\right)$. So, the equation now looks like: $\frac{1}{2} x+12-\frac{1}{2}\left(\frac{1}{3} x+14\right)-3$. Now, let's distribute the $-\frac{1}{2}$ to the terms inside the second set of parentheses: $-\frac{1}{2} \times \frac{1}{3} x$ and $-\frac{1}{2} \times 14$. For the first term, multiplying fractions means multiplying the numerators together and the denominators together: $-\frac{1 \times 1}{2 \times 3} x = -\frac{1}{6} x$. For the second term, $-\frac{1}{2} \times 14$ is equivalent to $-\frac{14}{2}$, which simplifies to $-7$. So, the expression inside the parentheses is now replaced by $-\frac{1}{6} x - 7$. Our equation is transforming into: $\frac{1}{2} x+12-\frac{1}{6} x-7-3$. This simplification process is fundamental in algebra. By carefully applying the distributive property, we eliminate the parentheses, making the equation much more manageable for the subsequent steps. It's important to pay close attention to the signs of each term as we distribute. For example, multiplying a negative number by a positive number results in a negative number, and multiplying two negative numbers results in a positive number. In our case, the negative sign in front of $\frac{1}{2}$ ensures that both terms inside the second parenthesis change their signs when distributed. This meticulous approach to simplification sets a strong foundation for accurately isolating x.

Step 2: Combine Like Terms

After simplifying the equation by distributing, our next objective in finding the value of x is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, $\frac{1}{2} x+12-\frac{1}{6} x-7-3$, the like terms involving x are $\frac{1}{2} x$ and $-\frac{1}{6} x$. The constant terms are $12$, $-7$, and $-3$. To combine the x terms, we need to find a common denominator for the fractions $\frac{1}{2}$ and $-\frac{1}{6}$. The least common denominator for 2 and 6 is 6. So, we can rewrite $\frac{1}{2}$ as $\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$. Now, we can combine the x terms: $\frac{3}{6} x - \frac{1}{6} x = \frac{3-1}{6} x = \frac{2}{6} x$. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, $\frac{2}{6} x$ simplifies to $\frac{1}{3} x$. Next, let's combine the constant terms: $12 - 7 - 3$. First, $12 - 7 = 5$. Then, $5 - 3 = 2$. So, the combined constant term is $2$. After combining like terms, our equation is now significantly simpler: $\frac{1}{3} x + 2$. This step is critical because it reduces the complexity of the equation, bringing us closer to isolating x. Combining like terms correctly ensures that we are not mixing different types of quantities, maintaining the integrity of our algebraic manipulations. The process of finding common denominators for fractions is a fundamental skill in arithmetic that extends directly into algebra, allowing us to perform operations on terms with fractional coefficients. Similarly, careful addition and subtraction of integers are essential. This consolidation of terms makes the equation much easier to solve in the final steps.

Step 3: Isolate x

We are now at the pivotal stage of solving for x, where we need to isolate x on one side of the equation. Our simplified equation is $\frac{1}{3} x + 2$. To isolate x, we first need to get rid of the constant term, which is $+2$. We do this by performing the opposite operation on both sides of the equation. The opposite of adding 2 is subtracting 2. So, we subtract 2 from both sides: $\frac{1}{3} x + 2 - 2 = 0 - 2$. This simplifies to $\frac{1}{3} x = -2$. Now, x is being multiplied by $\frac{1}{3}$. To isolate x, we need to perform the opposite operation of multiplying by $\frac{1}{3}$, which is dividing by $\frac{1}{3}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$ or simply 3. So, we multiply both sides of the equation by 3: $3 \times \frac{1}{3} x = 3 \times -2$. On the left side, $3 \times \frac{1}{3}$ cancels out to 1, leaving us with $1x$, or just $x$. On the right side, $3 \times -2 = -6$. Therefore, we have found that $x = -6$. This final step of isolating x involves inverse operations to undo the operations that are being applied to x. It's like unwrapping a present, carefully removing each layer until you get to the gift inside. By consistently applying the same operation to both sides of the equation, we maintain equality, ensuring that our solution is valid. The concept of inverse operations is a cornerstone of algebra, enabling us to solve for unknown variables in a systematic and reliable manner.

Conclusion: The Value of x

Through a series of carefully executed algebraic steps, we have successfully determined the value of x in the given equation. We began by simplifying the expression \frac{2}{2}\left(\frac{1}{2} x+12\right)-\frac{1}{2}\left(\frac{1}{3} x+14 ight)-3. This involved distributing the coefficients, which transformed the equation into $\frac{1}{2} x+12-\frac{1}{6} x-7-3$. Subsequently, we combined like terms, merging the x terms to get $\frac{1}{3} x$ and the constant terms to get $2$. This resulted in the much simpler equation: $\frac{1}{3} x + 2$. The final stage involved isolating x by using inverse operations. We subtracted 2 from both sides, leading to $\frac{1}{3} x = -2$, and then multiplied both sides by 3 to get $x = -6$. Thus, the value of x in the equation is -6. This systematic approach demonstrates the power of algebraic manipulation in solving for unknown variables. Each step, from simplification to combining like terms and finally isolating x, builds upon the last, leading to a clear and accurate solution. Understanding these principles not only helps in solving this specific problem but also equips you with the skills to tackle a wide range of algebraic challenges. For further exploration into algebraic equations and problem-solving techniques, you can refer to resources like Khan Academy. Their comprehensive lessons and practice exercises can deepen your understanding and proficiency in mathematics.