Solving For X In (2/3)x = -12 A Step-by-Step Guide

In the realm of algebra, solving for unknown variables is a fundamental skill. This article delves into the process of solving a linear equation, specifically $\frac{2}{3}x = -12$. We will explore the steps involved, providing a clear and concise explanation for both beginners and those seeking a refresher. This equation, (2/3)x = -12, is a straightforward example of a one-step linear equation, and mastering its solution lays the groundwork for tackling more complex algebraic problems. Let's embark on this mathematical journey together, demystifying the process and building a solid understanding of how to isolate the variable 'x'.

Understanding the Equation

Before diving into the solution, it's crucial to understand the structure of the equation (2/3)x = -12. This equation states that two-thirds of a number, represented by the variable 'x', is equal to -12. Our goal is to find the value of 'x' that makes this statement true. The equation is a linear equation because the variable 'x' is raised to the power of 1. Linear equations are the simplest type of algebraic equations, and they have a wide range of applications in various fields, from science and engineering to economics and finance. Understanding the basic principles of solving linear equations is therefore essential for anyone pursuing studies or careers in these areas. The coefficient of x is 2/3, which means 'x' is being multiplied by the fraction 2/3. To isolate 'x', we need to perform the inverse operation of multiplication, which is division. However, instead of directly dividing by 2/3, it's often easier to multiply by the reciprocal of 2/3. The reciprocal of a fraction is obtained by swapping its numerator and denominator. In this case, the reciprocal of 2/3 is 3/2. Multiplying both sides of the equation by 3/2 will effectively cancel out the fraction on the left side, leaving 'x' isolated. This is a key technique in solving linear equations, and it's important to grasp the underlying concept of inverse operations and reciprocals.

Step-by-Step Solution

To solve the equation (2/3)x = -12, we follow these steps:

1. Multiply both sides of the equation by the reciprocal of the coefficient of x. In this case, the coefficient of x is 2/3, and its reciprocal is 3/2. Multiplying both sides by 3/2 gives us: (3/2) * (2/3)x = (3/2) * (-12)

2. Simplify both sides of the equation. On the left side, the (3/2) and (2/3) cancel each other out, leaving us with just 'x'. On the right side, we multiply 3/2 by -12: x = (3/2) * (-12)

3. Perform the multiplication on the right side. To multiply a fraction by a whole number, we can rewrite the whole number as a fraction with a denominator of 1: x = (3/2) * (-12/1)

   Now, we multiply the numerators and the denominators: x = (3 * -12) / (2 * 1) x = -36 / 2

4. Simplify the fraction on the right side. Divide -36 by 2: x = -18

Therefore, the solution to the equation (2/3)x = -12 is x = -18. This step-by-step approach ensures clarity and accuracy in solving the equation. Each step is logically connected to the previous one, making the process easy to follow. By understanding the rationale behind each step, you can apply this method to solve similar linear equations with confidence. It's important to practice these steps with various examples to solidify your understanding and develop your problem-solving skills. Remember, algebra is a building block for more advanced mathematical concepts, and mastering these fundamental skills will pave the way for your future success in mathematics and related fields. The importance of carefully following each step cannot be overstated. A small mistake in any step can lead to an incorrect solution. Therefore, it's crucial to double-check your work and ensure that each calculation is accurate.

Verification

To ensure our solution is correct, we can substitute x = -18 back into the original equation (2/3)x = -12 and see if it holds true. Substituting x = -18, we get:

(2/3) * (-18) = -12

Multiply 2/3 by -18:

(2 * -18) / 3 = -12

-36 / 3 = -12

-12 = -12

Since the left side of the equation equals the right side, our solution x = -18 is correct. This verification step is a crucial part of the problem-solving process. It allows you to confirm the accuracy of your solution and catch any potential errors. By substituting the solution back into the original equation, you are essentially reversing the steps you took to solve the equation. If the equation holds true after the substitution, it provides strong evidence that your solution is correct. This practice not only ensures accuracy but also helps to reinforce your understanding of the equation and the solution process. Furthermore, verification can be particularly helpful when dealing with more complex equations where errors are more likely to occur. By developing the habit of verifying your solutions, you can build confidence in your problem-solving abilities and improve your overall mathematical skills. It is a valuable tool for both learning and assessment.

Alternative Methods

While multiplying by the reciprocal is the most common method for solving this type of equation, there's another way to approach it. Instead of multiplying by the reciprocal (3/2), we could first multiply both sides of the equation (2/3)x = -12 by 3. This would eliminate the denominator on the left side:

3 * (2/3)x = 3 * (-12)

2x = -36

Then, we would divide both sides by 2 to isolate 'x':

2x / 2 = -36 / 2

x = -18

This method achieves the same result, demonstrating that there can be multiple paths to the solution. Understanding alternative methods can be beneficial for several reasons. First, it provides you with more flexibility in solving equations. You can choose the method that you find most intuitive or efficient. Second, it deepens your understanding of the underlying mathematical principles. By exploring different approaches, you gain a more comprehensive perspective on the problem and the various ways to solve it. Third, it can help you to develop your problem-solving skills. The ability to think creatively and explore alternative solutions is a valuable asset in mathematics and beyond. In this particular case, multiplying by the denominator first and then dividing by the coefficient of 'x' is a perfectly valid and often preferred method by some students. It breaks down the problem into smaller, more manageable steps, which can make it easier to understand and solve. Ultimately, the best method is the one that you are most comfortable with and that you can apply consistently and accurately. The key is to understand the principles behind each method and to choose the one that works best for you.

Real-World Applications

Linear equations like (2/3)x = -12 might seem abstract, but they have numerous real-world applications. For example, imagine you're baking a cake and the recipe calls for two-thirds of a cup of flour per serving. If you want to make -12 servings (perhaps you're scaling down a recipe significantly, or the negative represents a deficit), you'd use this equation to find out how much flour you need. In this case, 'x' would represent the total amount of flour. Another example could involve calculating discounts. If an item is two-thirds of its original price and the discounted price is -12 dollars (representing a reduction), you could use this equation to find the original price. These practical applications highlight the importance of mastering algebraic skills. Linear equations are used extensively in various fields, including finance, engineering, and physics. They are the foundation for modeling real-world situations and solving problems that arise in these contexts. For instance, in finance, linear equations can be used to calculate interest rates, loan payments, and investment returns. In engineering, they can be used to design structures, analyze circuits, and control systems. In physics, they can be used to describe motion, forces, and energy. By understanding how to solve linear equations, you are equipping yourself with a valuable tool that can be applied to a wide range of real-world problems. Furthermore, the problem-solving skills you develop in algebra are transferable to other areas of life. The ability to analyze a problem, identify the key variables, and develop a logical solution is a valuable asset in any field. Therefore, mastering algebraic concepts is not only important for academic success but also for professional and personal growth.

Conclusion

Solving for 'x' in the equation (2/3)x = -12 is a fundamental algebraic exercise. By multiplying both sides by the reciprocal of the coefficient of 'x', we isolated 'x' and found that x = -18. We also verified our solution and explored an alternative method. Understanding the steps involved and practicing similar problems will solidify your algebraic skills. Remember, algebra is a building block for more advanced math, and mastering these basics is crucial for future success. The ability to solve linear equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. This article has provided a comprehensive guide to solving the equation (2/3)x = -12, covering the steps involved, verification, alternative methods, and real-world applications. By understanding the concepts and practicing the techniques discussed in this article, you can build a strong foundation in algebra and develop your problem-solving skills. Remember that mathematics is a cumulative subject, and each concept builds upon the previous one. Therefore, mastering the basics is essential for tackling more complex problems in the future. Continue to practice and explore different types of equations, and you will gradually develop your mathematical abilities and confidence. The journey of learning mathematics is a rewarding one, and the skills you acquire along the way will be valuable assets in your academic, professional, and personal life.