Solving For X: A Step-by-Step Guide

Let's dive into solving for x in the equation -30/x = -40/(x+2). This is a common type of algebraic problem that involves fractions and variables. Understanding how to solve these equations is crucial for many areas of mathematics and real-world applications. In this comprehensive guide, we'll break down the steps, explain the logic behind each one, and provide you with a clear understanding of how to tackle similar problems. This introduction sets the stage for a detailed exploration of solving equations with variables in the denominator. We'll cover everything from the basic principles to advanced techniques, ensuring you're well-equipped to handle any algebraic challenge that comes your way. So, grab your pencil and paper, and let's get started on this mathematical journey together! By the end of this article, you'll not only know how to solve this specific equation but also have a solid foundation for tackling a wide range of algebraic problems.

1. Understanding the Equation

Before we start manipulating the equation, it’s important to understand what it represents. We have an equation with two fractions, and our goal is to isolate x and find its value. The key here is to recognize that we can use algebraic manipulations to simplify the equation and eventually solve for x. This initial understanding is crucial because it guides our approach and helps us avoid common mistakes. Think of it like having a roadmap before embarking on a journey; it provides direction and ensures you reach your destination efficiently. Furthermore, understanding the equation involves recognizing any potential pitfalls, such as values of x that would make the denominator zero, which are not allowed. This step is not just about looking at the symbols; it’s about comprehending the relationship between the variables and the constants, setting the stage for a successful solution.

1.1. Identifying Restrictions on x

First, we need to identify any restrictions on x. Since we have x in the denominator, x cannot be 0. Additionally, x + 2 is in the denominator, so x + 2 cannot be 0, which means x cannot be -2. These restrictions are critical because they tell us which values of x are not valid solutions, regardless of what our algebraic manipulations might suggest. Failing to consider these restrictions can lead to incorrect answers or misunderstandings of the problem. It's like setting boundaries before playing a game; knowing the rules ensures fair play and prevents unnecessary errors. Therefore, before we proceed with solving the equation, we must keep in mind that x ≠ 0 and x ≠ -2. These conditions will be essential in verifying our final solution.

2. Eliminating the Fractions

The next step is to eliminate the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are x and (x + 2), so their LCM is x(x + 2). Multiplying both sides by this LCM will clear the fractions, making the equation easier to solve. This technique is a fundamental tool in algebra, allowing us to transform complex equations into simpler forms. Think of it like clearing debris from a path; it removes obstacles and allows for smoother progress. By eliminating fractions, we reduce the risk of errors and make the algebraic manipulations more straightforward. This step is crucial for solving equations involving rational expressions, and mastering it will significantly improve your problem-solving skills.

2.1. Multiplying by the LCM

Multiplying both sides of the equation by x(x + 2), we get:

x(x + 2) * (-30/x) = x(x + 2) * (-40/(x + 2)).

On the left side, the x cancels out, and on the right side, the (x + 2) cancels out. This leaves us with:

-30(x + 2) = -40x.

This is a significant simplification. We've transformed the original equation with fractions into a linear equation, which is much easier to handle. This process of multiplying by the LCM is like using a key to unlock a door; it opens up a new, more accessible path to the solution. The cancellation of terms is a direct result of the properties of multiplication and division, highlighting the elegance and efficiency of algebraic manipulations. By performing this step carefully, we've set the stage for solving the equation without the complexities of fractions, bringing us closer to finding the value of x.

3. Simplifying the Equation

Now, let’s simplify the equation. We start by distributing the -30 on the left side:

-30x - 60 = -40x.

This step involves applying the distributive property, a fundamental concept in algebra. It's like expanding a folded map; it reveals the details and makes the path clearer. Distributing the -30 allows us to remove the parentheses and combine like terms, which is essential for isolating x. By simplifying the equation, we reduce the complexity and make it easier to see the next steps. This process is akin to tidying up a workspace; it eliminates clutter and allows for more focused work. The careful application of the distributive property ensures that each term is correctly accounted for, paving the way for the subsequent steps in solving for x.

3.1. Combining Like Terms

Next, we want to isolate the x terms on one side of the equation. We can add 30x to both sides:

-60 = -40x + 30x.

This simplifies to:

-60 = -10x.

Combining like terms is a crucial step in solving equations. It’s like sorting puzzle pieces; it groups together elements that belong together, making the overall picture clearer. By adding 30x to both sides, we’ve effectively moved all the x terms to the right side, leaving a constant term on the left. This manipulation is based on the principle of maintaining equality; whatever operation we perform on one side of the equation, we must also perform on the other. The result is a much simpler equation, where the x term is isolated and ready for the final step of solving. This process of combining like terms is a fundamental skill in algebra, essential for simplifying equations and progressing towards the solution.

4. Solving for x

Finally, to solve for x, we divide both sides by -10:

x = -60 / -10.

This gives us:

x = 6.

Dividing both sides by -10 is the final step in isolating x and finding its value. It’s like using a magnifying glass to focus on the solution; it brings the answer into clear view. This step is based on the principle that dividing both sides of an equation by the same non-zero number maintains the equality. The result, x = 6, is the solution to the equation. However, it’s crucial to remember the restrictions we identified earlier. Since 6 is not 0 or -2, it is a valid solution. This final step of solving for x is the culmination of all the previous steps, highlighting the importance of each manipulation in reaching the correct answer. It’s a testament to the power of algebraic techniques in uncovering the hidden values of variables.

4.1. Checking the Solution

Before we declare victory, let's check our solution. Substitute x = 6 back into the original equation:

-30/6 = -40/(6 + 2).

This simplifies to:

-5 = -40/8.

And further to:

-5 = -5.

The equation holds true, so our solution x = 6 is correct. Checking the solution is a crucial step in the problem-solving process. It’s like proofreading a document; it catches any errors and ensures the final result is accurate. By substituting the value of x back into the original equation, we verify that it satisfies the equality. This step provides confidence in our solution and confirms that we have correctly applied the algebraic principles. In this case, the equality holds true, validating our solution x = 6. This final check reinforces the importance of thoroughness and attention to detail in mathematics, ensuring that we arrive at the correct answer with certainty.

5. Conclusion

In conclusion, we have successfully solved for x in the equation -30/x = -40/(x+2). The solution is x = 6. We achieved this by understanding the equation, eliminating fractions, simplifying the equation, and finally, isolating x. Remember to always check for restrictions on the variable and verify your solution. This step-by-step approach is applicable to a wide range of algebraic problems, making it a valuable tool in your mathematical toolkit. Mastering these techniques not only enhances your problem-solving skills but also provides a deeper understanding of algebraic principles. So, keep practicing, and you'll become more confident and proficient in solving equations. The journey through algebra is filled with challenges and rewards, and each solved problem brings you closer to mathematical mastery.

For further learning and practice on similar algebraic problems, you can explore resources like Khan Academy, which offers a wealth of educational materials and exercises.