Solving 8x-2(x-5)=x+25: A Step-by-Step Guide

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the world of mathematics to tackle a classic algebraic equation: 8x - 2(x - 5) = x + 25. If you've ever found yourself staring at a jumble of numbers and variables, wondering where to even begin, you're in the right place. We're going to break this down, piece by piece, so that by the end, you'll not only understand how to solve this particular equation but also gain the confidence to tackle others like it. Mathematics can seem intimidating, but trust me, with a little bit of logic and a systematic approach, it becomes surprisingly manageable, and dare I say, even fun!

So, let's get straight to it. Our mission is to find the value of 'x' that makes the equation 8x - 2(x - 5) = x + 25 true. Think of it like a balancing scale; whatever you do to one side, you must do to the other to keep it balanced. Our goal is to isolate 'x' on one side of the equation. We'll start by simplifying both sides of the equation. The left side, 8x - 2(x - 5), has parentheses, which means we need to distribute that '-2' to both terms inside the parentheses. This is a common stumbling block for many, so pay close attention here. When we multiply '-2' by 'x', we get '-2x'. When we multiply '-2' by '-5', remember that a negative times a negative equals a positive, so we get '+10'. So, the left side now transforms into 8x - 2x + 10. Before we combine terms, let's just re-state the equation with this simplification: 8x - 2x + 10 = x + 25. Now, we can combine the 'x' terms on the left side. We have '8x' and we're subtracting '2x', which leaves us with 6x. So, the equation is now 6x + 10 = x + 25. See? We're already making progress and simplifying the whole thing. This initial step of distribution and combining like terms is crucial for setting us up for the next stages of solving.

Simplifying and Isolating 'x'

Alright, fam, we've simplified the left side of our equation 8x - 2(x - 5) = x + 25 to 6x + 10 = x + 25. Now, the game plan is to get all the 'x' terms on one side of the equation and all the constant numbers on the other. This is where the balancing act really comes into play. Let's decide to move the 'x' terms to the left side. To do this, we need to eliminate the '+x' from the right side. The opposite of adding 'x' is subtracting 'x', so we'll subtract 'x' from both sides of the equation. On the right side, 'x - x' becomes 0, leaving us with just '25'. On the left side, we have '6x - x', which simplifies to 5x. So now, our equation looks like this: 5x + 10 = 25. We're one step closer to getting 'x' all by itself! Keep your eyes on the prize, guys.

Next, we need to move the constant term, '+10', from the left side to the right side. To get rid of that '+10', we perform the opposite operation: we subtract 10 from both sides of the equation. On the left side, '+10 - 10' cancels out to 0, leaving us with just 5x. On the right side, we have '25 - 10', which equals 15. So, the equation is now a beautiful, simple 5x = 15. Can you feel the victory? We've successfully isolated the term containing 'x'. This process of moving terms by using inverse operations is the backbone of solving algebraic equations. It ensures that the equality of the equation is maintained throughout the process, no matter how many steps it takes. Remember, every step is about undoing what's being done to 'x' in a way that keeps the equation balanced.

The Final Step: Finding the Value of 'x'

We're at the finish line, people! Our equation has been simplified down to 5x = 15. This literally means '5 multiplied by x equals 15'. To find the value of a single 'x', we need to undo that multiplication by 5. The inverse operation of multiplication is division. So, we're going to divide both sides of the equation by 5. On the left side, '5x divided by 5' simplifies to just x. And on the right side, '15 divided by 5' equals 3. So, there you have it: x = 3. We have successfully solved the equation! Isn't that awesome? This final step is always about isolating 'x' by performing the inverse operation of whatever is directly attached to it. If it were 'x/5', we'd multiply by 5. If it were 'x+5', we'd subtract 5, and so on. The key is to always do the opposite to maintain the balance of the equation.

Now, for the ultimate test: checking our answer. This is a super important step that many forget, but it's your best friend in proving you're right. We substitute our solution, x = 3, back into the original equation: 8x - 2(x - 5) = x + 25. Let's plug in 3 for every 'x':

Left Side: 8(3) - 2(3 - 5) Right Side: (3) + 25

Let's calculate the left side: 8(3) = 24 (3 - 5) = -2 -2(-2) = 4 So, the left side is 24 + 4 = 28.

Now, let's calculate the right side: 3 + 25 = 28.

Since the left side (28) equals the right side (28), our solution x = 3 is correct! This check confirms that our hard work paid off. It’s a fantastic feeling of accomplishment when you see both sides match up. This verification step is not just for this equation; it's a universal technique that you can apply to any equation you solve. It builds confidence and ensures accuracy, making you a more capable problem-solver. So, always make time for the check – it’s worth it!

We've now successfully navigated the equation 8x - 2(x - 5) = x + 25, from simplifying the parentheses to isolating 'x' and finally verifying our answer. Remember, the core principles are distribution, combining like terms, using inverse operations to move terms across the equals sign, and finally, checking your work. These techniques are fundamental building blocks in algebra, and the more you practice them, the more intuitive they become. Mathematics is a journey of building skills, and each equation you solve is a step forward. Don't be afraid to practice with different equations; the more you do, the more comfortable you'll become with manipulating variables and numbers. Keep experimenting, keep learning, and most importantly, keep having fun with it! Until next time, stay curious and keep those mathematical gears turning here at Plastik Magazine!