Solving For U: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of algebra to tackle a common problem: solving for a variable. In this case, we're going to break down how to solve the equation (5/6)u + 2 - u = (1/6)u. Don't worry if it looks intimidating at first. We'll take it one step at a time, and you'll be a pro in no time!
Understanding the Equation
Before we jump into solving, let's make sure we understand what the equation is telling us. We have an equation with the variable 'u' on both sides. Our goal is to isolate 'u' on one side of the equation so we can determine its value. This involves using algebraic manipulations to simplify the equation and get 'u' by itself.
Remember, the key principle in solving equations is to maintain balance. Whatever operation we perform on one side of the equation, we must also perform on the other side to keep the equation true. This is like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it balanced.
Now, let's get started with the first step in solving for 'u'. We'll focus on combining like terms, which will help us simplify the equation and make it easier to work with. Think of it as organizing your toolbox before you start a project – it makes the whole process smoother!
Step 1: Combine Like Terms
The first thing we want to do is simplify both sides of the equation by combining like terms. On the left side, we have two terms with 'u': (5/6)u and -u. Remember that -u is the same as -1u. To combine these, we need a common denominator. Let's rewrite -1u as (-6/6)u. Now we can add the coefficients:
(5/6)u + (-6/6)u = -1/6u
So, the left side of the equation simplifies to -1/6u + 2. The right side of the equation, (1/6)u, remains the same for now. Our equation now looks like this:
-1/6u + 2 = 1/6u
Combining like terms is a fundamental step in solving algebraic equations. It helps to tidy up the equation and make it more manageable. By grouping similar terms together, we reduce the number of individual components we need to deal with, making the subsequent steps clearer and easier to execute. Think of it as decluttering your workspace before starting a project – a clean space leads to a clearer mind!
Now that we've combined the 'u' terms on the left side, we're ready to move on to the next step: isolating the 'u' terms on one side of the equation. This will bring us closer to our goal of solving for 'u'.
Step 2: Isolate the 'u' Terms
Our next goal is to get all the 'u' terms on one side of the equation and the constant terms (the numbers without 'u') on the other side. To do this, we can add (1/6)u to both sides of the equation. This will eliminate the 'u' term on the left side:
-1/6u + 2 + 1/6u = 1/6u + 1/6u
Simplifying this, we get:
2 = 2/6u
Notice that we've successfully moved all the 'u' terms to the right side of the equation, leaving the constant term '2' on the left side. This is a crucial step in isolating the variable we're trying to solve for.
The principle of isolating variables is a cornerstone of algebra. It allows us to separate the unknown quantity from the known quantities, making it possible to determine its value. By strategically adding or subtracting terms from both sides of the equation, we can rearrange the equation to our advantage.
Now that we have the 'u' term isolated on one side, we're just one step away from finding the value of 'u'. We need to get rid of the coefficient (the number in front of 'u') to finally solve for 'u'. Let's move on to the final step!
Step 3: Solve for 'u'
We're almost there! Our equation is now 2 = (2/6)u. To solve for 'u', we need to get rid of the fraction (2/6) that's multiplying 'u'. We can do this by multiplying both sides of the equation by the reciprocal of 2/6, which is 6/2 (or simply 3):
2 * (6/2) = (2/6)u * (6/2)
Simplifying this, we get:
6 = u
And there you have it! We've solved for 'u'. The solution to the equation is u = 6.
To verify our solution, we can substitute u = 6 back into the original equation and see if it holds true. This is a good practice to ensure we haven't made any mistakes along the way.
(5/6)(6) + 2 - 6 = (1/6)(6)
5 + 2 - 6 = 1
1 = 1
The equation holds true, so we know our solution u = 6 is correct!
Solving for a variable involves a series of steps, each building upon the previous one. By understanding the underlying principles and practicing regularly, you'll become more confident and proficient in tackling algebraic equations. Remember, the key is to break down the problem into smaller, manageable steps and to maintain balance by performing the same operations on both sides of the equation.
Key Takeaways
Let's recap the key steps we took to solve for 'u':
- Combine Like Terms: Simplify both sides of the equation by combining terms that have the same variable or are constants.
- Isolate the 'u' Terms: Use addition or subtraction to get all the 'u' terms on one side of the equation and the constant terms on the other side.
- Solve for 'u': Multiply or divide both sides of the equation by the coefficient of 'u' to isolate 'u' and find its value.
Solving for variables is a fundamental skill in algebra and mathematics in general. It's a skill that you'll use repeatedly in more advanced topics, so mastering it now will set you up for success in the future. Think of it as building a strong foundation for your mathematical journey!
Practice Makes Perfect
The best way to get comfortable with solving equations is to practice. Try solving similar equations on your own. You can find practice problems in textbooks, online resources, or even create your own! The more you practice, the more natural the process will become.
Remember, guys, mathematics is like a muscle – the more you exercise it, the stronger it gets. Don't be afraid to make mistakes; they are part of the learning process. Each time you solve a problem, you're building your understanding and confidence.
If you get stuck on a problem, don't hesitate to seek help. Ask your teacher, a classmate, or look for online resources. There are many people and tools available to support your learning journey.
Conclusion
So, there you have it! We've successfully solved for 'u' in the equation (5/6)u + 2 - u = (1/6)u. By following these steps and practicing regularly, you'll be able to tackle similar problems with ease. Remember to combine like terms, isolate the variable, and don't be afraid to ask for help when you need it.
Solving algebraic equations is an essential skill in mathematics, and with practice, you'll become more confident and proficient. Keep practicing, and you'll be amazed at what you can achieve!
Keep up the great work, guys, and happy solving!