Solving For U A Step-by-Step Guide To (7/5)u - 5 = -(6/5)u - (2/3)
#Introduction In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving for the variable 'u' in the equation (7/5)u - 5 = -(6/5)u - (2/3). We will provide a step-by-step solution, ensuring clarity and understanding for readers of all levels. Understanding how to solve such equations is crucial for various mathematical and scientific applications. It builds a foundation for more complex problem-solving scenarios and enhances analytical thinking. This article aims to break down the equation into manageable steps, making the solution accessible and the process understandable.
Understanding the Equation
Before diving into the solution, it is important to understand the structure of the equation. The equation (7/5)u - 5 = -(6/5)u - (2/3) is a linear equation in one variable, 'u'. Linear equations are equations in which the highest power of the variable is 1. This type of equation can be represented in the form ax + b = cx + d, where a, b, c, and d are constants. In our equation, the terms involve fractions and negative signs, which might appear daunting at first. However, by systematically applying the rules of algebra, we can simplify and solve for 'u'. The key is to isolate 'u' on one side of the equation, which will give us its value. Understanding the basics of linear equations is crucial for successfully solving this type of problem.
Step-by-Step Solution
1. Clearing the Fractions
The first step in solving the equation is to eliminate the fractions. This can be done by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 5 and 3, so the LCM is 15. Multiplying both sides of the equation by 15 gives:
15 * [(7/5)u - 5] = 15 * [-(6/5)u - (2/3)]
Distributing the 15 on both sides, we get:
(15 * 7/5)u - (15 * 5) = -(15 * 6/5)u - (15 * 2/3)
Simplifying each term:
21u - 75 = -18u - 10
By multiplying through by the LCM, we have transformed the equation into a more manageable form without fractions. This step is critical for simplifying the equation and making it easier to solve.
2. Grouping the 'u' Terms
The next step is to group all the terms containing 'u' on one side of the equation and the constants on the other side. To do this, we can add 18u to both sides of the equation:
21u - 75 + 18u = -18u - 10 + 18u
This simplifies to:
39u - 75 = -10
Now, we have all the 'u' terms on the left side of the equation. Grouping similar terms is an essential algebraic technique that simplifies the process of solving for a variable.
3. Isolating the 'u' Term
To further isolate 'u', we need to get rid of the constant term (-75) on the left side. We can do this by adding 75 to both sides of the equation:
39u - 75 + 75 = -10 + 75
This simplifies to:
39u = 65
Now, we have the 'u' term isolated on one side of the equation. Isolating the variable is a fundamental step in solving any equation.
4. Solving for 'u'
Finally, to solve for 'u', we need to divide both sides of the equation by the coefficient of 'u', which is 39:
39u / 39 = 65 / 39
This gives us:
u = 65/39
5. Simplifying the Fraction
The fraction 65/39 can be simplified. Both 65 and 39 are divisible by 13. Dividing both the numerator and the denominator by 13, we get:
u = (65 / 13) / (39 / 13)
u = 5/3
Therefore, the solution to the equation is u = 5/3. Simplifying the answer is a crucial step in ensuring that the solution is in its simplest form.
Alternative Methods for Solving the Equation
While the step-by-step method described above is a standard approach, there are alternative ways to solve the equation. One such method involves rearranging the terms before clearing the fractions. This can sometimes simplify the process, depending on the specific equation. Another method involves using a calculator or computer algebra system (CAS) to solve the equation. These tools can quickly find the solution, but it is important to understand the underlying steps involved in solving the equation manually. Exploring alternative methods can enhance problem-solving skills and provide different perspectives on the same problem.
Common Mistakes to Avoid
When solving equations, it is important to avoid common mistakes that can lead to incorrect answers. One common mistake is incorrectly distributing the multiplier when clearing fractions. Another mistake is making errors when adding or subtracting terms on both sides of the equation. It is also important to simplify the final answer as much as possible. Being aware of common mistakes can help in avoiding them and improving accuracy in solving equations.
Practical Applications of Solving Equations
Solving equations is a fundamental skill in mathematics and has numerous practical applications in various fields. In physics, equations are used to describe the motion of objects and the interactions between them. In engineering, equations are used to design structures, circuits, and other systems. In economics, equations are used to model economic phenomena and make predictions. Solving equations is also important in everyday life, such as when calculating finances or planning a budget. Understanding the practical applications of solving equations can motivate learners and make the topic more engaging.
Tips for Mastering Equation Solving
Mastering equation solving requires practice and a solid understanding of the underlying concepts. Here are some tips to help you improve your skills: 1. Practice regularly: The more you practice, the better you will become at solving equations. 2. Understand the concepts: Make sure you understand the basic principles of algebra, such as the order of operations and the properties of equality. 3. Break down the problem: Complex equations can be broken down into smaller, more manageable steps. 4. Check your work: Always check your answer to make sure it is correct. 5. Seek help when needed: If you are struggling with a particular type of equation, don't hesitate to ask for help from a teacher, tutor, or online resource. Mastering equation solving is a continuous process that requires effort and dedication.
Conclusion
In conclusion, solving for 'u' in the equation (7/5)u - 5 = -(6/5)u - (2/3) involves several steps, including clearing fractions, grouping terms, isolating the variable, and simplifying the answer. By following these steps carefully, you can successfully solve the equation and arrive at the solution u = 5/3. Solving equations is a fundamental skill in mathematics and has numerous practical applications in various fields. Mastering this skill requires practice, a solid understanding of the underlying concepts, and attention to detail. This article has provided a comprehensive guide to solving this particular equation, but the principles and techniques discussed can be applied to a wide range of similar problems. Mastering equation solving not only enhances mathematical skills but also cultivates critical thinking and problem-solving abilities.
