Solving For U In -(5/3)u = 15 A Step-by-Step Guide

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Introduction

In mathematics, solving for variables is a fundamental skill. This article provides a detailed guide on how to solve for u in the equation -(5/3)u = 15. We will break down the steps, explain the underlying principles, and offer insights to ensure you understand the process thoroughly. Mastering this type of equation is crucial for more advanced mathematical concepts. Before diving into the solution, it’s essential to grasp the basic algebraic principles that govern equation-solving. The main goal is to isolate the variable u on one side of the equation. To achieve this, we use inverse operations, which are operations that undo each other. For example, multiplication is the inverse of division, and addition is the inverse of subtraction. Understanding these concepts will make the process of solving equations much more intuitive. Let's embark on this mathematical journey and unravel the solution step by step.

Understanding the Equation

The equation we aim to solve is -(5/3)u = 15. This is a linear equation, which means it involves a variable (u) raised to the power of 1. The coefficient of u is -(5/3), which is a fraction. The right-hand side of the equation is a constant, 15. To solve for u, we need to isolate it on one side of the equation. This involves performing operations on both sides of the equation to maintain equality. The key concept here is the multiplicative inverse. The multiplicative inverse of a number is a value which, when multiplied by the original number, results in 1. For example, the multiplicative inverse of 2 is 1/2, and the multiplicative inverse of -(5/3) is -(3/5). Using the multiplicative inverse is the most efficient way to isolate u in this equation. Keeping the balance of the equation is paramount; any operation performed on one side must also be performed on the other side. This principle ensures that the solution remains accurate and the equality holds true throughout the solving process. With a clear understanding of the equation and the principles involved, we are now ready to move on to the step-by-step solution.

Step-by-Step Solution

To solve the equation -(5/3)u = 15, we need to isolate u. The most direct method is to multiply both sides of the equation by the multiplicative inverse of -(5/3), which is -(3/5). Here’s how it looks:

  1. Multiply both sides by -(3/5): -(3/5) * (-(5/3)u) = -(3/5) * 15
  2. Simplify the left side: The negative signs cancel out, and (3/5) * (5/3) equals 1, so we have: 1 * u = -(3/5) * 15 u = -(3/5) * 15
  3. Simplify the right side: Multiply -(3/5) by 15. We can think of 15 as 15/1, so we have: u = -(3 * 15) / (5 * 1) u = -45 / 5
  4. Divide -45 by 5: u = -9

Therefore, the solution for u is -9. Each step in this process is crucial. Multiplying by the multiplicative inverse efficiently cancels out the coefficient of u, leaving u isolated. The simplification steps ensure that the equation is manageable and easy to compute. By following these steps meticulously, we arrive at the correct solution. Now that we have the solution, it is important to verify it to ensure its accuracy.

Verifying the Solution

To ensure our solution u = -9 is correct, we substitute it back into the original equation -(5/3)u = 15. This process of verification is a critical step in solving equations, as it helps to identify any potential errors in the solution. Here’s how we verify:

  1. Substitute u = -9 into the original equation: -(5/3) * (-9) = 15
  2. Simplify the left side: Multiply -(5/3) by -9. We can think of -9 as -9/1, so we have: (-5 * -9) / (3 * 1) = 15 45 / 3 = 15
  3. Divide 45 by 3: 15 = 15

Since the left side equals the right side, our solution u = -9 is correct. This verification step confirms that we have accurately solved the equation. It is always a good practice to verify solutions, especially in exams or when dealing with complex equations. Verification not only confirms the correctness of the solution but also reinforces the understanding of the equation-solving process. By substituting the solution back into the original equation, we ensure that the equality holds true, thereby validating our answer.

Common Mistakes to Avoid

When solving equations like -(5/3)u = 15, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One frequent error is forgetting to apply the operation to both sides of the equation. For example, if you multiply the left side by -(3/5) to isolate u, you must also multiply the right side by the same value. Failing to do so will disrupt the equality and lead to an incorrect solution. Another common mistake is mishandling the negative signs. In this particular equation, the negative sign in front of (5/3)u needs to be carefully considered. Students might drop the negative sign or make errors in multiplying negative numbers. Remember that a negative times a negative is a positive, and a negative times a positive is a negative. It is also crucial to simplify fractions correctly. When multiplying or dividing fractions, ensure that you simplify the result to its lowest terms. For instance, if you end up with 45/5, simplify it to 9. Neglecting to simplify fractions can lead to confusion and increase the chances of making errors in subsequent steps. Finally, always double-check your work. After you have found a solution, take the time to substitute it back into the original equation to verify its correctness. This step can catch simple arithmetic errors or mistakes in applying operations. By being mindful of these common pitfalls and taking steps to avoid them, you can improve your accuracy and confidence in solving algebraic equations.

Alternative Methods

While multiplying by the multiplicative inverse is the most efficient method for solving -(5/3)u = 15, there are alternative approaches that can also lead to the correct solution. Understanding these methods can broaden your problem-solving skills and offer different perspectives on equation solving. One alternative method is to first multiply both sides of the equation by the denominator of the fraction, which in this case is 3. This step eliminates the fraction and simplifies the equation. Here’s how it works:

  1. Multiply both sides by 3: 3 * (-(5/3)u) = 3 * 15
  2. Simplify: -5u = 45

Now, the equation is -5u = 45. To isolate u, you would then divide both sides by -5:

  1. Divide both sides by -5: (-5u) / -5 = 45 / -5
  2. Simplify: u = -9

This method avoids dealing with fractions directly in the first step and can be easier for some students to grasp. Another approach involves dividing both sides of the equation by -(5/3). Dividing by a fraction is the same as multiplying by its reciprocal, which is exactly what we did in the step-by-step solution. However, some students might find it more intuitive to think of it as division. These alternative methods illustrate that there are often multiple paths to the same solution in mathematics. The key is to choose the method that you understand best and can apply accurately. By exploring different techniques, you can enhance your problem-solving abilities and develop a deeper understanding of algebraic principles.

Real-World Applications

Understanding how to solve equations like -(5/3)u = 15 is not just a matter of academic exercise; it has numerous real-world applications across various fields. Algebraic equations are the backbone of many problem-solving scenarios in science, engineering, economics, and everyday life. In physics, for example, equations are used to describe motion, forces, and energy. If you need to calculate the force required to move an object with a certain acceleration, you might encounter an equation similar to the one we solved. Similarly, in engineering, equations are used to design structures, circuits, and systems. Solving for variables helps engineers determine the optimal parameters for their designs, ensuring efficiency and safety. Economics also relies heavily on algebraic equations to model markets, predict trends, and make financial decisions. For instance, if you are calculating the return on an investment or determining the break-even point for a business, you will likely use equations to solve for unknown variables. Even in everyday situations, solving equations comes in handy. For example, if you are trying to budget your expenses, calculate discounts while shopping, or determine the amount of ingredients needed for a recipe based on the number of servings, you are essentially solving equations. The ability to manipulate and solve algebraic equations is a valuable skill that empowers you to tackle a wide range of problems effectively. By mastering these fundamental concepts, you are better equipped to navigate and understand the world around you.

Conclusion

In this comprehensive guide, we have thoroughly explored how to solve the equation -(5/3)u = 15. We began by understanding the equation and the basic algebraic principles involved, emphasizing the importance of inverse operations and the multiplicative inverse. We then walked through a step-by-step solution, demonstrating how to isolate u by multiplying both sides of the equation by -(3/5). The solution, u = -9, was carefully verified by substituting it back into the original equation, confirming its accuracy. We also discussed common mistakes to avoid, such as failing to apply operations to both sides of the equation or mishandling negative signs. Alternative methods for solving the equation were presented, illustrating that there are often multiple paths to the same correct answer. Finally, we highlighted the real-world applications of solving equations, emphasizing its relevance in fields such as science, engineering, economics, and everyday life. Mastering the ability to solve algebraic equations is a fundamental skill that opens doors to a deeper understanding of mathematics and its applications. By practicing these techniques and understanding the underlying principles, you can build confidence in your problem-solving abilities and tackle more complex mathematical challenges. Remember, the key to success in mathematics is consistent practice and a solid grasp of the fundamentals. With dedication and the right approach, you can excel in solving equations and unlock the power of mathematics.