Solving For X A Step-by-Step Guide To Making X The Subject

In mathematics, rearranging formulas to isolate a specific variable, often called making a variable the subject of the formula, is a fundamental skill. This process allows us to solve for that variable if we know the values of the other variables in the equation. In this comprehensive guide, we'll walk through the steps involved in making $x$ the subject of the formula in the equation $6\left(\frac{x}{c}-p\right)=h$. By the end of this article, you'll not only understand the mechanics of this particular problem but also grasp the underlying principles that can be applied to a wide range of similar algebraic manipulations. We will break down each step with detailed explanations and examples, ensuring clarity and building a solid foundation for more advanced algebraic concepts. Mastering this technique is crucial for various mathematical and scientific applications, where you often need to express one variable in terms of others. From physics equations to economic models, the ability to rearrange formulas is an indispensable tool for problem-solving and analysis. Let's delve into the process of isolating $x$ in the given equation, unraveling the steps with precision and care.

Understanding the Basics of Subject Manipulation

Before we dive into the specific equation, it's crucial to understand the foundational principles of subject manipulation. At its core, making a variable the subject involves using inverse operations to undo the operations that are applied to the variable we want to isolate. This means if a variable is being multiplied by a number, we divide by that number. If it's being added to a term, we subtract that term, and so on. The golden rule in this process is that whatever operation you perform on one side of the equation, you must perform on the other side to maintain the balance and ensure the equation remains true. Think of an equation as a weighing scale; to keep it balanced, any weight you add or remove from one side must be added or removed from the other. This principle is the cornerstone of algebraic manipulation and is essential for solving equations accurately. Understanding this concept allows you to approach equation solving with confidence and precision. It’s not just about following steps mechanically; it’s about understanding why each step is necessary to maintain the integrity of the equation. This foundational knowledge will empower you to tackle more complex algebraic problems and apply these techniques in various mathematical contexts. So, with this fundamental understanding, let’s move on to the specifics of our problem and see how these principles are applied in practice.

Step-by-Step Solution: Isolating $x$

Now, let's tackle the given equation step-by-step. Our goal is to isolate $x$ on one side of the equation. The equation we're working with is $6\left(\frac{x}{c}-p\right)=h$.

Step 1: Distribute the 6

Our first step is to eliminate the parentheses by distributing the 6 across the terms inside. This means we multiply both $\frac{x}{c}$ and $-p$ by 6. This gives us: $6 \cdot \frac{x}{c} - 6 \cdot p = h$, which simplifies to $\frac{6x}{c} - 6p = h$. This step is crucial because it removes the grouping symbol, allowing us to work with individual terms more easily. Distributing the constant factor helps to unravel the equation and prepare it for subsequent steps in isolating $x$. By applying the distributive property, we transform the equation into a more manageable form, setting the stage for the next operations. This foundational algebraic technique is vital for simplifying expressions and solving equations efficiently. So, having distributed the 6, we’ve made significant progress in our journey to make $x$ the subject of the formula.

Step 2: Add $6p$ to Both Sides

Next, we want to isolate the term containing $x$. To do this, we need to get rid of the $-6p$ term on the left side of the equation. We accomplish this by adding $6p$ to both sides. This maintains the balance of the equation and isolates the term with $x$. The equation now looks like this: $\frac{6x}{c} - 6p + 6p = h + 6p$. Simplifying this, we get $\frac{6x}{c} = h + 6p$. Adding $6p$ to both sides is a critical step in isolating the term containing $x$. It demonstrates the fundamental principle of maintaining equality in an equation by performing the same operation on both sides. This step brings us closer to our goal of making $x$ the subject of the formula. By strategically adding $6p$, we’ve effectively removed the constant term from the left side, paving the way for further manipulation and eventual isolation of $x$.

Step 3: Multiply Both Sides by $c$

Now we have $\frac{6x}{c} = h + 6p$. To further isolate $x$, we need to eliminate the denominator $c$. We do this by multiplying both sides of the equation by $c$. This gives us $c \cdot \frac{6x}{c} = c \cdot (h + 6p)$, which simplifies to $6x = c(h + 6p)$. Multiplying both sides by $c$ is a crucial step as it removes the fraction, making the equation easier to manipulate. This action brings us closer to isolating $x$ by undoing the division operation. It’s another clear demonstration of how inverse operations are used to rearrange equations. By multiplying by $c$, we’ve cleared the denominator and simplified the equation significantly, setting the stage for the final step in making $x$ the subject of the formula. This process showcases the power of algebraic manipulation in simplifying complex expressions and solving for unknown variables.

Step 4: Divide Both Sides by 6

Finally, we have $6x = c(h + 6p)$. To completely isolate $x$, we need to get rid of the coefficient 6. We do this by dividing both sides of the equation by 6. This gives us $\frac{6x}{6} = \frac{c(h + 6p)}{6}$, which simplifies to $x = \frac{c(h + 6p)}{6}$. This final step completes the process of making $x$ the subject of the formula. Dividing both sides by 6 isolates $x$, providing us with the solution. This action demonstrates the use of the inverse operation to undo multiplication and is the culmination of our step-by-step manipulation. With this step, we have successfully rearranged the equation to express $x$ in terms of the other variables. The final equation, $x = \frac{c(h + 6p)}{6}$, clearly shows $x$ as the subject, and we have achieved our objective through careful and methodical algebraic manipulation.

The Final Result

Therefore, making $x$ the subject of the formula $6\left(\frac{x}{c}-p\right)=h$ gives us $x = \frac{c(h + 6p)}{6}$. This result allows us to easily calculate the value of $x$ if we know the values of $c$, $h$, and $p$. The process we've followed illustrates a fundamental skill in algebra: rearranging equations to solve for a specific variable. This skill is not only crucial for solving mathematical problems but also has wide applications in various fields, including physics, engineering, and economics. Understanding how to manipulate equations empowers you to solve problems efficiently and accurately. By mastering these techniques, you can confidently tackle more complex algebraic challenges and apply your knowledge in practical scenarios. The ability to make a variable the subject of a formula is a cornerstone of mathematical literacy, opening doors to further learning and problem-solving in diverse disciplines.

Common Mistakes to Avoid

When rearranging formulas, it's easy to make mistakes if you're not careful. One common mistake is not applying the same operation to both sides of the equation, which can lead to an imbalance and an incorrect result. Remember, the principle of equality dictates that any operation performed on one side must be mirrored on the other. Another frequent error is incorrectly applying the order of operations. For example, failing to distribute a factor across terms inside parentheses before attempting other operations can lead to errors. Always follow the correct order of operations (PEMDAS/BODMAS) to ensure accuracy. Additionally, sign errors are a common pitfall, particularly when dealing with negative numbers. Double-check your work when adding, subtracting, multiplying, or dividing negative terms to avoid mistakes. Another error occurs when students fail to simplify the equation after each step, leading to more complex expressions and increasing the likelihood of mistakes. Simplifying as you go helps keep the equation manageable and reduces the chances of error. Lastly, forgetting to isolate the variable completely is a common oversight. Ensure that the variable you're making the subject is the only term on one side of the equation. By being mindful of these common mistakes and double-checking your work, you can improve your accuracy and confidence in rearranging formulas.

Practice Problems

To solidify your understanding of making a variable the subject of a formula, practice is essential. Here are a few problems you can try:

1. Make $y$ the subject of $3(y + 2z) = x$
2. Make $a$ the subject of $\frac{4a}{b} - 5c = d$
3. Make $r$ the subject of $7(r - 3q) = p$

Working through these problems will help reinforce the steps and principles we've discussed. Each problem offers a unique challenge and provides an opportunity to apply the techniques you've learned. Start by identifying the variable you need to isolate and then methodically apply inverse operations to both sides of the equation. Remember to distribute any factors, combine like terms, and simplify the equation at each step. Don't be afraid to double-check your work and review the steps if you encounter any difficulties. The key to mastering this skill is consistent practice and careful attention to detail. By tackling these practice problems, you'll not only improve your algebraic proficiency but also build confidence in your problem-solving abilities. So, grab a pen and paper, and start practicing to become proficient in making variables the subject of formulas.

Conclusion

In conclusion, making $x$ the subject of the formula $6\left(\frac{x}{c}-p\right)=h$ is a valuable exercise in algebraic manipulation. The step-by-step process involves distributing, adding, multiplying, and dividing to isolate the variable. By understanding and applying these principles, you can confidently rearrange equations and solve for any variable. Mastering this skill is not only crucial for success in mathematics but also for a wide range of applications in science, engineering, and other fields. Remember, the key to proficiency is practice, so keep working on rearranging formulas to build your skills and confidence. Algebraic manipulation is a fundamental tool in problem-solving, and the ability to make a variable the subject of a formula is a cornerstone of algebraic competence. With a solid understanding of the principles and consistent practice, you can tackle more complex equations and confidently apply these skills in various contexts. So, continue to hone your algebraic skills, and you'll be well-equipped to solve a wide range of mathematical and real-world problems.