Solving For U In 28 = -6u - 12 + 2u A Step-by-Step Guide

Hey guys! Today, we're diving into a fundamental algebraic problem: solving for the variable u in the equation 28 = -6u - 12 + 2u. If you're just starting out with algebra or need a refresher, you've come to the right place! We'll break down each step, making it super easy to follow along. Let's get started and unravel this equation together!

Understanding the Basics of Algebraic Equations

Before we jump into the solution, let's quickly recap what an algebraic equation is all about. In simple terms, an equation is a mathematical statement that shows the equality between two expressions. Our main goal when solving an equation is to isolate the variable (in this case, u) on one side of the equation. This means we want to get u by itself so that we can determine its value. To do this, we use various algebraic operations while maintaining the balance of the equation – what we do on one side, we must also do on the other. Think of it like a see-saw; if you add or subtract weight on one side, you need to do the same on the other to keep it balanced.

In our equation, 28 = -6u - 12 + 2u, we have a linear equation, which means the highest power of the variable u is 1. These types of equations are common and relatively straightforward to solve. The key is to follow the order of operations (PEMDAS/BODMAS) in reverse when isolating the variable. We'll be combining like terms, using inverse operations, and simplifying to get to our final answer. So, with a clear understanding of these basics, let’s move on to the first step in solving for u. Remember, the goal is to make the process as clear and understandable as possible, so don’t hesitate to rewind or re-read if needed. Let's get into the nitty-gritty of solving this equation!

Step 1: Combining Like Terms

Alright, the first thing we need to do when tackling our equation, 28 = -6u - 12 + 2u, is to combine like terms. What does this mean? Well, like terms are terms that have the same variable raised to the same power. In our equation, we have two terms that contain the variable u: -6u and +2u. These guys are like terms because they both have u raised to the power of 1. The other term, -12, is a constant term, meaning it doesn't have a variable attached to it. We can't combine it with the terms that have u.

To combine -6u and +2u, we simply add their coefficients. The coefficient is the number that multiplies the variable. So, we have -6 + 2, which equals -4. This means that -6u + 2u simplifies to -4u. Our equation now looks like this: 28 = -4u - 12. See how much cleaner that looks? Combining like terms is a fundamental step in solving algebraic equations because it simplifies the equation, making it easier to work with. It's like tidying up your workspace before starting a project – it just makes everything smoother!

Now that we've combined the like terms, our equation is more manageable. We've reduced the number of terms on the right side, bringing us closer to isolating u. This step is crucial because it sets the stage for the next operations we'll perform. So, remember, whenever you see an equation with multiple terms involving the same variable, your first move should always be to combine those like terms. It's a simple yet powerful technique that will make your algebraic journey much easier. Next up, we'll be moving the constant term to the other side of the equation. Stay tuned!

Step 2: Isolating the Variable Term

Okay, we've combined like terms and our equation now stands as 28 = -4u - 12. The next step in our mission to solve for u is to isolate the variable term. In this case, the variable term is -4u. Remember, our goal is to get u all by itself on one side of the equation. To do this, we need to get rid of the -12 that's hanging out on the same side as -4u.

How do we do that? We use the magic of inverse operations! Since 12 is being subtracted from -4u, we need to do the opposite operation, which is addition. We're going to add 12 to both sides of the equation. Why both sides? Because, as we discussed earlier, we need to maintain the balance of the equation. What we do on one side, we must do on the other. So, let's add 12 to both sides:

28 + 12 = -4u - 12 + 12

On the left side, 28 + 12 equals 40. On the right side, -12 + 12 cancels out, leaving us with just -4u. Our equation now looks like this: 40 = -4u. We're getting closer! Notice how adding 12 to both sides helped us to isolate the term with u. This is a key technique in solving equations, and you'll use it time and time again. By strategically using inverse operations, we're peeling away the layers of the equation to reveal the value of u.

So, we've successfully isolated the variable term -4u. We're now one step closer to finding the value of u. The next step involves dealing with the coefficient that's multiplying u. We'll continue using inverse operations to completely isolate u. Keep going, guys! We're almost there!

Step 3: Solving for u

Alright, we've reached the final stretch! Our equation is currently 40 = -4u. We've isolated the variable term, but u is not completely alone yet. It's being multiplied by -4. To finally solve for u, we need to get rid of this -4. And you guessed it – we'll use inverse operations again!

Since u is being multiplied by -4, the inverse operation is division. We're going to divide both sides of the equation by -4. Remember, we must do the same thing to both sides to keep the equation balanced. So, let's divide both sides by -4:

40 / -4 = -4u / -4

On the left side, 40 divided by -4 is -10. On the right side, -4u divided by -4 simplifies to just u. So, our equation now looks like this: -10 = u. Ta-da! We've done it! We've solved for u. This means that the value of u that makes the original equation true is -10.

This final step is where all our hard work pays off. By using the inverse operation of division, we've successfully isolated u and found its value. This is the essence of solving algebraic equations – carefully applying operations to both sides until the variable is all alone. So, the solution to our equation 28 = -6u - 12 + 2u is u = -10. We've taken a complex-looking equation and broken it down into manageable steps, ultimately arriving at the answer. Great job, guys!

Checking Our Solution

Before we celebrate our victory, it's always a good idea to double-check our answer. This is a crucial step in solving any equation because it helps us catch any potential errors. To check our solution, we'll substitute the value we found for u (which is -10) back into the original equation. If both sides of the equation are equal after the substitution, then we know our solution is correct.

Our original equation was 28 = -6u - 12 + 2u. Let's replace u with -10:

28 = -6(-10) - 12 + 2(-10)

Now, we simplify each side of the equation using the order of operations (PEMDAS/BODMAS). First, we perform the multiplications:

28 = 60 - 12 - 20

Next, we perform the subtractions from left to right:

28 = 48 - 20

28 = 28

Look at that! Both sides of the equation are equal. This confirms that our solution, u = -10, is indeed correct. Checking our solution is like the final seal of approval on our work. It gives us confidence that we've solved the equation accurately and haven't made any mistakes along the way.

So, remember, always take the time to check your solutions. It's a simple step that can save you from making errors and ensure you're on the right track. By substituting our value back into the original equation, we've verified that our solution is correct. Awesome work, guys!

Conclusion: Mastering the Art of Solving Equations

And there you have it! We've successfully solved for u in the equation 28 = -6u - 12 + 2u. We started by combining like terms, then isolated the variable term, and finally solved for u using inverse operations. We even took the extra step to check our solution, ensuring its accuracy. This whole process is a fantastic example of how algebraic equations can be broken down into manageable steps.

Solving equations is a fundamental skill in mathematics, and it's something you'll use again and again in various contexts. Whether you're dealing with simple linear equations or more complex problems, the core principles remain the same: combine like terms, use inverse operations to isolate the variable, and always check your solution. The more you practice, the more comfortable and confident you'll become with these techniques.

Remember, algebra is like a puzzle, and solving equations is like finding the missing piece. Each step we take brings us closer to the solution, and the satisfaction of cracking the code is truly rewarding. So, don't be intimidated by equations. Embrace the challenge, break them down into steps, and remember the tools we've discussed today. You've got this!

I hope this comprehensive guide has been helpful in your algebraic journey. Keep practicing, keep exploring, and most importantly, keep having fun with math! Until next time, guys! Keep solving!