Solve $3(2x+6)-4x=2(5x-2)+6$

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the world of mathematics to tackle a problem that might look a little intimidating at first glance: solving the equation $3(2x+6)-4x=2(5x-2)+6$. Don't worry, though! By the end of this article, you'll be a pro at simplifying and finding the value of 'x' in no time. We'll break down each step, making sure you understand the logic behind it. So grab your thinking caps, and let's get started on this algebraic adventure!

Understanding the Equation: A First Look

Alright, let's eyeball this equation: $3(2x+6)-4x=2(5x-2)+6$. Our main mission, should we choose to accept it (and we totally should!), is to find the value of 'x' that makes this equation true. Think of 'x' as a mystery number that we need to uncover. To do that, we need to isolate 'x' on one side of the equals sign. This involves a few key algebraic maneuvers. First up, we need to simplify both sides of the equation. That means dealing with those pesky parentheses. Remember the distributive property? It's our best friend here. It tells us that when we have a number multiplied by a term in parentheses, we multiply that number by each term inside the parentheses. So, for the left side, $3(2x+6)$, we'll multiply 3 by $2x$ and then multiply 3 by 6. For the right side, $2(5x-2)$, we'll multiply 2 by $5x$ and then multiply 2 by -2. This will get rid of the parentheses and give us a simpler expression to work with. It's like unwrapping a present – once you get rid of the packaging, you can see what's inside more clearly. This initial simplification is crucial because it lays the groundwork for all the subsequent steps. Without properly distributing, the entire solution could be thrown off. So, take your time, double-check your multiplication, and make sure those signs are correct, especially when dealing with subtraction. This foundational step is where many people stumble, so let's give it the attention it deserves. We're not just blindly applying rules; we're strategically manipulating the equation to get closer to our goal of finding 'x'.

Step 1: Distribute and Simplify the Left Side

Okay, team, let's tackle the left side of our equation: $3(2x+6)-4x$. We're going to use that distributive property I mentioned. This means we multiply the 3 by everything inside the parentheses. So, $3 \times 2x$ gives us $6x$, and $3 \times 6$ gives us $18$. Now, our left side looks like $6x + 18 - 4x$. We're not done simplifying yet! See those 'x' terms? We have $6x$ and $-4x$. These are like terms, meaning they both have 'x' in them. We can combine them. $6x - 4x$ equals $2x$. So, the entire left side simplifies to $2x + 18$. Awesome job, guys! This step is all about cleaning up one side of the equation before we even touch the other. It makes the equation much more manageable. Think of it as tidying up your workspace before you start a big project. The more organized you are, the smoother the process will be. When distributing, always be mindful of the signs. If there were a negative sign in front of the 3, for example, we'd have to multiply by a negative, which would flip the signs of the terms inside the parentheses. In this case, it's all positive, which makes it a bit easier. Combining like terms is another fundamental skill. You can only add or subtract terms that have the same variable raised to the same power. Here, both $6x$ and $-4x$ are 'x' to the first power, so they're good to go. The $+18$ is a constant term, so it stays separate. So, from $3(2x+6)-4x$, we've successfully transformed it into $2x+18$. This is a significant reduction in complexity, and it shows we're making real progress towards isolating 'x'. Remember this simplified form, as it's what we'll be working with going forward on the left side of our original equation.

Step 2: Distribute and Simplify the Right Side

Now, let's move over to the right side of the equation: $2(5x-2)+6$. Again, we hit those parentheses, so the distributive property is called into action. We multiply the 2 by everything inside the parentheses. $2 \times 5x$ is $10x$, and $2 \times -2$ is $-4$. So, the right side becomes $10x - 4 + 6$. Just like before, we need to combine any like terms. Here, we have two constant terms: $-4$ and $+6$. When we combine them, $-4 + 6$ equals $2$. So, the entire right side simplifies to $10x + 2$. We're making great progress, everyone! We've now simplified both sides of the original equation. This is a huge milestone. The process here mirrors the left side: identify the terms within parentheses, apply the distributive property, and then combine any remaining like terms. The key here is the $-2$ inside the parentheses. When multiplying $2 \times -2$, we get $-4$. It's crucial to keep track of those negative signs. Then, combining $-4$ with the $+6$ outside the parentheses gives us a positive $2$. This is why carefully performing each arithmetic operation is so important. A single slip-up can lead to a completely different answer. So, the original right side, $2(5x-2)+6$, has now been neatly condensed into $10x+2$. This is our simplified form for the right side. We're now in a position to bring the simplified left and right sides together and solve for 'x'. High five! This systematic approach ensures that we're not missing any steps and that our calculations are as accurate as possible. We've taken two complex expressions and made them much easier to handle.

Step 3: Combine Simplified Sides

Alright, folks, we've done the hard work of simplifying both sides. Let's put them back together. Our original equation was $3(2x+6)-4x=2(5x-2)+6$. After our brilliant simplification, the left side became $2x + 18$, and the right side became $10x + 2$. So, our new, much friendlier equation is: $2x + 18 = 10x + 2$. Now, the goal is to get all the 'x' terms on one side and all the constant terms on the other. It doesn't matter which side you choose for the 'x' terms, but it's often easier to move the smaller 'x' term to avoid dealing with negative coefficients for 'x' later. In this case, $2x$ is smaller than $10x$. So, let's subtract $2x$ from both sides of the equation to keep it balanced. On the left side, $2x - 2x$ cancels out, leaving us with just $18$. On the right side, $10x - 2x$ gives us $8x$. So now our equation is $18 = 8x + 2$. Feeling good? We're getting so close! This step is all about consolidation. We've taken the simplified expressions and are now setting them equal to each other. The strategy of moving the smaller 'x' term is a smart move because it means the coefficient of 'x' on the remaining side will be positive. If we had subtracted $10x$ from both sides, we would have had $-8x$ on the left, and then we'd have to divide by a negative number later, which is perfectly fine, but sometimes an extra step where errors can creep in. By subtracting $2x$, we maintain a positive $8x$. Remember, whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side to maintain the equality. This is the golden rule of algebra: keep it balanced! So, from $2x + 18 = 10x + 2$, we've arrived at $18 = 8x + 2$. The 'x' terms are now consolidated on the right side, and we just need to move the constant term over there too.

Step 4: Isolate the Constant Terms

We're on the home stretch, folks! Our equation is currently $18 = 8x + 2$. We want to get the $8x$ term all by itself on the right side. To do that, we need to get rid of the $+2$. The opposite of adding 2 is subtracting 2. So, let's subtract 2 from both sides of the equation. On the right side, $8x + 2 - 2$ leaves us with just $8x$. On the left side, $18 - 2$ gives us $16$. So, our equation is now $16 = 8x$. We're literally one step away from the answer! This step is all about isolating the term containing our variable. We've moved all the constant terms to the side opposite the variable terms. By subtracting 2 from both sides, we've effectively undone the $+2$ that was on the right side, leaving only the term with 'x'. This maintains the balance of the equation. Imagine a scale; if you remove weight from one side, you must remove the same weight from the other to keep it level. Here, we're removing the value of 2 from both sides. So, the equation $18 = 8x + 2$ transforms into $16 = 8x$. This simplified form clearly shows the relationship between the number 16 and the term $8x$. Our next and final step will be to determine the value of a single 'x'. We've done the heavy lifting, and now it's just a matter of a final division.

Step 5: Solve for 'x'

We've arrived at the grand finale! Our equation is $16 = 8x$. Remember, this means 16 is equal to 8 times x. To find out what x is, we need to do the opposite of multiplying by 8, which is dividing by 8. So, we divide both sides of the equation by 8. On the right side, $8x / 8$ leaves us with just $x$. On the left side, $16 / 8$ equals $2$. Therefore, $x = 2$! We did it! We've successfully solved the equation. The mystery number 'x' is 2. Give yourselves a massive round of applause! This final step is the payoff for all the meticulous work we've done. We've successfully isolated the variable 'x' and now we just need to find its value. Since $8x$ means $8$ multiplied by $x$, the inverse operation is division. By dividing both sides by 8, we effectively cancel out the coefficient of $x$, leaving $x$ by itself. The result of $16$ divided by $8$ is $2$. So, our solution is $x=2$. To be absolutely sure, you can always plug this value back into the original equation to check if it holds true. Let's do that quickly: $3(2(2)+6)-4(2) = 2(5(2)-2)+6$. Simplifying the left side: $3(4+6)-8 = 3(10)-8 = 30-8 = 22$. Simplifying the right side: $2(10-2)+6 = 2(8)+6 = 16+6 = 22$. Since both sides equal 22, our solution $x=2$ is indeed correct! It's always a good habit to check your work, especially in math. It builds confidence and ensures accuracy. You guys absolutely crushed this!

Conclusion: You've Mastered the Equation!

And there you have it, math enthusiasts! We've journeyed through the steps of solving the equation $3(2x+6)-4x=2(5x-2)+6$, and the answer is a resounding $x = 2$. We started by bravely facing the parentheses, using the distributive property to simplify both sides. Then, we masterfully combined like terms, setting up a cleaner equation. The key steps involved isolating the 'x' terms on one side and the constant terms on the other, all while maintaining the crucial balance of the equation. Finally, a simple division gave us our solution. Remember, guys, the process of solving algebraic equations is all about systematic simplification and strategic manipulation. Each step builds upon the last, leading you closer to the answer. Don't be intimidated by complex-looking equations; break them down, tackle them piece by piece, and you'll find they're much more manageable than you think. Keep practicing, keep exploring, and you'll become an algebra whiz in no time. Thanks for joining me on Plastik Magazine for this mathematical dive! Until next time, stay curious and keep those brains sharp!