Solving The Equation 12(x+5) = 4x: A Step-by-Step Guide

Hey math enthusiasts! Ever stumbled upon an equation that looks a bit intimidating at first glance? Don't worry, we've all been there. Today, we're going to break down a common type of algebraic equation and show you how to solve it step-by-step. We'll be tackling the equation 12(x + 5) = 4x. This might seem tricky, but with a few simple rules, you'll be solving these like a pro in no time. So, let's dive in and make math a little less mysterious, shall we?

Understanding the Equation

Before we jump into the solution, let's make sure we understand what the equation is asking. The equation 12(x + 5) = 4x is an algebraic equation, which means we're trying to find the value of the unknown variable, 'x', that makes the equation true. In simpler terms, we need to figure out what number 'x' represents so that when we plug it into the equation, both sides of the equals sign are the same. This involves using the principles of algebra to isolate 'x' on one side of the equation. Understanding this fundamental concept is crucial because it sets the stage for the entire solving process. We're not just randomly moving numbers around; we're strategically manipulating the equation to reveal the hidden value of 'x'. To solve this, we'll need to use a combination of the distributive property, combining like terms, and isolating the variable. Each of these steps plays a vital role in unraveling the equation and getting us closer to the solution. So, keep this in mind as we proceed – we're on a mission to find 'x', and we'll use all the tools at our disposal to get there! Moreover, remember that equations are like a balanced scale. Whatever you do to one side, you must do to the other to maintain the balance. This principle is fundamental in algebra and will guide our steps as we manipulate the equation. So, let's keep this balance in mind as we move forward, ensuring that each step we take brings us closer to the correct solution while maintaining the integrity of the equation.

Step 1: Apply the Distributive Property

The first step in solving our equation is to get rid of the parentheses. Remember the distributive property? It's our trusty tool for multiplying a number by a sum or difference inside parentheses. In our case, we have 12(x + 5). This means we need to multiply 12 by both 'x' and 5. So, 12 multiplied by 'x' is simply 12x, and 12 multiplied by 5 is 60. Therefore, 12(x + 5) becomes 12x + 60. Now, our equation looks like this: 12x + 60 = 4x. See? We've already made progress! Applying the distributive property is often the first step in simplifying equations with parentheses, and it's a crucial skill to master. By distributing the 12, we've transformed the equation into a more manageable form, setting the stage for further simplification. It's like peeling back the first layer of an onion – we're getting closer to the core of the problem. This step not only simplifies the equation but also allows us to see the individual terms more clearly, making it easier to proceed with the next steps. So, remember the distributive property – it's your friend in simplifying algebraic expressions!

Step 2: Combine Like Terms

Now that we've distributed, we need to gather our 'x' terms on one side of the equation. We have 12x + 60 = 4x. To get all the 'x' terms together, let's subtract 4x from both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced! Subtracting 4x from both sides gives us: 12x - 4x + 60 = 4x - 4x. Simplifying this, we get 8x + 60 = 0. Awesome! We've successfully moved the 'x' terms to one side. Combining like terms is a fundamental technique in algebra, and it helps us simplify the equation by grouping similar terms together. In this case, we grouped the 'x' terms, but we could also group constant terms (numbers without variables). This process makes the equation cleaner and easier to work with, bringing us closer to isolating the variable. By subtracting 4x from both sides, we've essentially canceled out the 4x on the right side, leaving us with only the constant term. This is a strategic move that helps us isolate the 'x' term on the left side, which is our ultimate goal. So, remember, combining like terms is a powerful tool for simplifying equations and making them more manageable. It's like organizing your workspace before tackling a big project – it makes the task much easier!

Step 3: Isolate the Variable

We're getting closer! Our equation is now 8x + 60 = 0. To isolate 'x', we need to get rid of the +60. How do we do that? You guessed it – we subtract 60 from both sides of the equation. This gives us: 8x + 60 - 60 = 0 - 60, which simplifies to 8x = -60. Almost there! Isolating the variable is the key to solving any algebraic equation. It's like peeling away the layers of an onion until you get to the core. In this case, we're peeling away the constant term (+60) to reveal the 'x' term. By subtracting 60 from both sides, we maintain the balance of the equation while moving closer to our goal of isolating 'x'. This step is crucial because it sets us up for the final step, where we'll divide to solve for 'x'. So, remember, isolating the variable is a fundamental skill in algebra, and it involves using inverse operations (like subtraction to undo addition) to get the variable by itself on one side of the equation. It's like a game of strategic moves, where each step brings you closer to the solution!

Step 4: Solve for x

We've got 8x = -60. Now, to finally solve for 'x', we need to get rid of the 8 that's multiplying it. To do this, we'll divide both sides of the equation by 8. This gives us: 8x / 8 = -60 / 8, which simplifies to x = -7.5. Ta-da! We've found our solution. Solving for 'x' is the grand finale of our algebraic journey. It's the moment where all our hard work pays off and we discover the value of the unknown variable. In this case, we divided both sides of the equation by 8 to isolate 'x', which is the inverse operation of multiplication. This step is like the final piece of the puzzle falling into place, revealing the complete picture. The solution, x = -7.5, is the value that makes the original equation true. If we were to plug -7.5 back into the equation 12(x + 5) = 4x, both sides would be equal. So, congratulations! You've successfully solved the equation. Remember, solving for 'x' is the ultimate goal in many algebraic problems, and it involves using inverse operations to isolate the variable. It's a rewarding feeling when you finally arrive at the solution, knowing that you've conquered the equation!

Checking Your Answer (Optional but Recommended)

To be absolutely sure we've got it right, it's always a good idea to check our answer. We can do this by plugging our solution, x = -7.5, back into the original equation: 12(x + 5) = 4x. Let's substitute -7.5 for 'x': 12(-7.5 + 5) = 4(-7.5). Now, let's simplify: 12(-2.5) = -30 and -30 = -30. Hooray! Both sides of the equation are equal, so our solution is correct. Checking your answer is like having a safety net in algebra. It's a quick and easy way to ensure that you haven't made any mistakes along the way. By plugging your solution back into the original equation, you can verify that both sides are equal, confirming that your answer is correct. This step is especially helpful in catching any arithmetic errors or sign mistakes that might have occurred during the solving process. It's also a great way to build confidence in your problem-solving skills. Knowing that you've checked your answer and it's correct can give you a sense of accomplishment and encourage you to tackle more challenging problems. So, remember, checking your answer is an optional but highly recommended step in algebra. It's like double-checking your work before submitting it – it ensures accuracy and gives you peace of mind!

Conclusion

And there you have it! We've successfully solved the equation 12(x + 5) = 4x, and found that x = -7.5. Remember, solving equations is like a puzzle – each step gets you closer to the final answer. By following these steps – applying the distributive property, combining like terms, isolating the variable, and solving for 'x' – you can tackle all sorts of algebraic equations. Keep practicing, and you'll become a math whiz in no time! So, the next time you encounter an equation that seems daunting, remember the steps we've discussed, and approach it with confidence. Math might seem intimidating at times, but with the right tools and techniques, you can conquer any problem. And who knows, you might even start to enjoy the challenge! Remember, every equation solved is a step forward in your mathematical journey. So, keep practicing, keep learning, and keep exploring the fascinating world of mathematics!

So, guys, wasn't that a fun little math adventure? We took a potentially scary equation and broke it down into simple, manageable steps. Now you've got another tool in your math belt! Keep practicing, and you'll be solving equations like a total boss. Until next time, keep those brains buzzing! And remember, math isn't just about numbers and symbols; it's about problem-solving, logical thinking, and the thrill of finding the right answer. So, embrace the challenge, and don't be afraid to make mistakes along the way. Every mistake is a learning opportunity, and with each equation you solve, you're building your confidence and sharpening your skills. So, go forth and conquer those mathematical mountains! You've got this!