Solving (3/4)x = -6 In One Step A Comprehensive Guide
In this article, we will delve into the process of solving the equation $ rac{3}{4}x = -6$ in a single step. This is a fundamental concept in algebra, and understanding it thoroughly is crucial for tackling more complex equations. We'll explore the underlying principles, discuss why certain methods work, and provide a step-by-step explanation to ensure clarity. This guide aims to equip you with the knowledge and confidence to solve similar equations efficiently and accurately. Mastering this skill will significantly enhance your problem-solving abilities in mathematics and related fields. Our focus will be on identifying the correct operation that isolates the variable x on one side of the equation. The key to solving such equations lies in understanding the concept of inverse operations. Each mathematical operation has an inverse that undoes it. For example, the inverse of multiplication is division, and vice versa. Similarly, the inverse of addition is subtraction, and vice versa. When solving equations, our goal is to isolate the variable by performing the inverse operation on both sides of the equation. This maintains the balance of the equation while simplifying it. In the given equation, $ rac{3}{4}x = -6$, the variable x is being multiplied by the fraction $ rac{3}{4}$. To isolate x, we need to perform the inverse operation of multiplication, which is division. However, instead of dividing by a fraction, which can be cumbersome, we multiply by the reciprocal of the fraction. The reciprocal of $ rac{3}{4}$ is $ rac{4}{3}$. Multiplying a fraction by its reciprocal results in 1, effectively canceling out the fraction and isolating the variable. This technique is a cornerstone of algebraic manipulation and is widely used in solving various types of equations. In the following sections, we will dissect the given options and demonstrate why multiplying by the reciprocal is the most efficient method for solving this particular equation in one step. We will also highlight the importance of maintaining the equality by performing the same operation on both sides of the equation. This fundamental principle ensures that the solution obtained is accurate and valid. By the end of this guide, you will have a firm grasp of this technique and be able to apply it to solve a wide range of similar equations with confidence. This skill is not only essential for academic success but also for various real-world applications where algebraic problem-solving is required.
Understanding the Equation (3/4)x = -6
Before diving into the solution, let's first understand the equation $ rac{3}{4}x = -6$. This equation states that three-fourths of a certain number (x) is equal to -6. Our objective is to find the value of x that satisfies this condition. To do this, we need to isolate x on one side of the equation. This involves performing operations that effectively “undo” the operations currently acting on x. In this case, x is being multiplied by the fraction $ rac{3}{4}$. The inverse operation of multiplication is division, but as mentioned earlier, we can achieve the same result more efficiently by multiplying by the reciprocal of the fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of $ rac{3}{4}$ is $ rac{4}{3}$. Multiplying both sides of the equation by $ rac{4}{3}$ will cancel out the $ rac{3}{4}$ on the left side, leaving us with x isolated. This is the core principle behind solving this type of equation in one step. It's crucial to remember that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the equality. This ensures that the equation remains balanced and the solution we obtain is correct. Failing to do so would lead to an incorrect value for x. The concept of reciprocals and their use in solving equations is a fundamental building block in algebra. It's a technique that is used extensively in more advanced mathematical concepts and problem-solving scenarios. Therefore, a solid understanding of this principle is essential for anyone pursuing further studies in mathematics or related fields. In the subsequent sections, we will analyze the given options and demonstrate how multiplying by the reciprocal leads to the correct solution in a single step. We will also discuss why the other options are not the most efficient or correct approaches. By carefully examining each option, we will reinforce your understanding of the principles involved and equip you with the ability to solve similar equations with greater confidence and accuracy.
Analyzing Option A: (4/3)(3/4)x = -6(4/3)
Option A presents the solution as $ rac4}{3}\left(\frac{3}{4}\right) x=-6\left(\frac{4}{3}\right)$. This option correctly identifies the key concept of multiplying by the reciprocal. The reciprocal of $ rac{3}{4}$ is indeed $ rac{4}{3}$, and this option demonstrates multiplying both sides of the equation by this reciprocal. On the left side of the equation, multiplying $ rac{4}{3}$ by $ rac{3}{4}x$ effectively cancels out the fraction $ rac{3}{4}$, leaving x isolated. This is because $ rac{4}{3}$ multiplied by $ rac{3}{4}$ equals 1. So, the left side simplifies to 1*x, which is simply x. On the right side of the equation, we have -6 multiplied by $ rac{4}{3}$. This is a straightforward multiplication of a whole number by a fraction. To perform this multiplication, we can rewrite -6 as a fraction, which is $ rac{-6}{1}$. Then, we multiply the numerators and the denominators{1} \times \frac{4}{3} = \frac{-6 \times 4}{1 \times 3} = \frac{-24}{3}$. Simplifying the fraction $ rac{-24}{3}$ gives us -8. Therefore, the right side of the equation simplifies to -8. Putting it all together, the equation becomes x = -8. This is the correct solution to the equation $ rac{3}{4}x = -6$. Option A demonstrates the most efficient way to solve the equation in one step. It correctly applies the principle of multiplying by the reciprocal and maintains the equality of the equation by performing the same operation on both sides. This method is a fundamental technique in algebra and is widely used in solving various types of equations. Understanding why this method works is crucial for developing strong algebraic problem-solving skills. In the following sections, we will compare this option with other given options and explain why they are not as efficient or correct. By analyzing the different approaches, we will further solidify your understanding of the principles involved and equip you with the ability to choose the most appropriate method for solving similar equations in the future. This comprehensive approach will enhance your overall mathematical proficiency and confidence.
Why Option A is the Correct One-Step Solution
Option A, as we've established, is the correct method to solve the equation $ rac{3}{4}x = -6$ in a single step. The reason lies in the fundamental principle of inverse operations. To isolate x, we need to undo the multiplication by $ rac{3}{4}$. The inverse operation of multiplying by a fraction is multiplying by its reciprocal. The reciprocal of $ rac{3}{4}$ is $ rac{4}{3}$. By multiplying both sides of the equation by $ rac{4}{3}$, we effectively cancel out the fraction on the left side, leaving x by itself. This is the most direct and efficient way to solve the equation. The alternative approaches might eventually lead to the correct answer, but they involve more steps and are therefore less efficient. For instance, one might consider dividing both sides by $ rac{3}{4}$, which is mathematically equivalent to multiplying by $ rac{4}{3}$. However, multiplying by the reciprocal is generally considered a more straightforward approach, especially when dealing with fractions. It avoids the potential confusion that can arise from dividing by a fraction. Another reason why Option A is the best choice is its clarity. It clearly demonstrates the application of the reciprocal property, which is a core concept in algebra. This makes the solution easy to understand and remember. By mastering this technique, students can confidently solve a wide range of similar equations with ease. Furthermore, the one-step approach minimizes the chances of making errors. Each additional step in a solution introduces the possibility of a calculation mistake. By solving the equation in one step, we reduce this risk and increase the likelihood of obtaining the correct answer. In the following sections, we will examine the other options and explain why they are not the most efficient or correct solutions. This comparative analysis will further reinforce your understanding of the principles involved and help you develop critical thinking skills in problem-solving. By the end of this discussion, you will have a comprehensive understanding of the best approach to solve equations of this type and be able to apply this knowledge with confidence and accuracy.
Analyzing Option B: 4(3/4)x = -6(4)
Option B presents the solution as $4\left(\frac{3}{4}\right) x=-6(4)$. While this option involves multiplying both sides of the equation by 4, it's not the most efficient way to isolate x in one step. Multiplying by 4 only addresses the denominator of the fraction $ rac{3}{4}$, but it doesn't deal with the numerator. This means that after this step, the equation will be $3x = -24$, which then requires an additional step to divide both sides by 3 to finally solve for x. In contrast, Option A, by multiplying by the reciprocal $ rac{4}{3}$, eliminates the fraction in one go, directly isolating x. Option B, therefore, is a multi-step solution disguised as a single step. It's important to recognize that the goal is not just to manipulate the equation but to do so in the most efficient way possible. In this case, multiplying by 4 introduces an unnecessary step. Furthermore, multiplying by 4 can sometimes lead to larger numbers, which can increase the chance of making calculation errors. While the arithmetic in this specific example is relatively simple, in more complex equations, larger numbers can make the problem more challenging. It's always a good practice to look for the most streamlined approach to minimize the risk of errors and save time. The key takeaway here is that while Option B is a valid algebraic manipulation, it's not the most efficient way to solve the equation in one step. It's crucial to understand the underlying principles of solving equations and to choose the method that directly addresses the operation acting on the variable. In this case, multiplying by the reciprocal is the most direct and efficient way to isolate x. In the following sections, we will further reinforce this concept by comparing Option B with Option A and highlighting the advantages of the one-step reciprocal approach. This comparative analysis will solidify your understanding of the principles involved and equip you with the ability to choose the most appropriate method for solving similar equations in the future.
Why Option B Requires an Additional Step
To further clarify why Option B isn't the ideal one-step solution, let's break down what happens after the multiplication. The equation becomes $4\left(\frac3}{4}\right) x=-6(4)$. Simplifying the left side, we have $3x = -24$. Now, to isolate x, we need to perform another operation{3}$. The extra step in Option B not only makes the solution process longer but also introduces an additional opportunity for errors. Each time you perform a calculation, there's a chance of making a mistake. By minimizing the number of steps, you reduce the risk of errors and increase the efficiency of your problem-solving. Furthermore, understanding the concept of inverse operations is crucial for mastering algebra. Option A directly applies this concept by using the reciprocal to undo the multiplication by $\frac{3}{4}$. Option B, on the other hand, only partially applies this concept, requiring an additional step to complete the isolation of x. Therefore, choosing Option A not only solves the equation more efficiently but also reinforces a fundamental algebraic principle. This understanding will be invaluable as you tackle more complex equations and mathematical problems in the future. In the subsequent sections, we will summarize the key takeaways from this analysis and emphasize the importance of choosing the most efficient method for solving equations. By consistently applying these principles, you will develop strong problem-solving skills and gain confidence in your ability to tackle a wide range of mathematical challenges.
Conclusion: Choosing the Right Approach
In conclusion, when solving the equation $ rac{3}{4}x = -6$, Option A, which demonstrates multiplying both sides by the reciprocal $ rac{4}{3}$, is the correct one-step solution. This approach directly isolates x by undoing the multiplication by $ rac{3}{4}$. Option B, while a valid algebraic manipulation, requires an additional step and is therefore less efficient. The key takeaway is that understanding and applying the concept of inverse operations, particularly the use of reciprocals, is crucial for solving equations efficiently and accurately. By choosing the most direct method, you minimize the chances of errors and save time. This principle applies not only to this specific equation but to a wide range of algebraic problems. Mastering the technique of multiplying by the reciprocal is a valuable skill that will serve you well in your mathematical journey. It's important to remember that the goal is not just to find the correct answer but also to develop a deep understanding of the underlying principles. This understanding will empower you to tackle more complex problems with confidence and creativity. As you continue to practice and apply these concepts, you will develop strong problem-solving skills and a solid foundation in algebra. This foundation will be invaluable for your future studies in mathematics and related fields. In the next steps of your learning, consider exploring more complex equations involving fractions and other operations. Challenge yourself to identify the most efficient method for solving each equation and to explain why that method is the best choice. By consistently applying these principles, you will strengthen your skills and deepen your understanding of algebra. Remember, mathematics is not just about memorizing formulas and procedures; it's about developing logical thinking and problem-solving abilities. By focusing on understanding the underlying principles, you will unlock the true power of mathematics and its applications in the world around you.
