Simple Math: Solve 5(x-1)+3x=7(x+1)

Hey math whizzes and everyone curious about cracking algebraic codes! Today, we're diving deep into a super common type of math puzzle: solving linear equations. Specifically, we're tackling the beast that is $5(x-1)+3x=7(x+1)$. Don't let those parentheses and multiple 'x' terms scare you off, guys. We're going to break it down step-by-step, making it as clear as mud (just kidding, as clear as crystal!). This equation, at its core, is asking us to find the unknown value of 'x' that makes the entire statement true. Think of it like a balanced scale; whatever you do to one side, you must do to the other to keep it level. Our mission, should we choose to accept it, is to isolate 'x' on one side of the equals sign. This might seem like a straightforward task, but it requires a solid understanding of fundamental algebraic principles. We'll be using properties like the distributive property, combining like terms, and the golden rule of algebra: doing the same operation to both sides of the equation. By the end of this, you'll not only have the solution but also a reinforced understanding of how to tackle similar problems. So, grab your favorite beverage, get comfy, and let's unravel this algebraic mystery together! We're aiming to demystify the process, making it accessible and even a little bit fun. Remember, every equation is just a puzzle waiting to be solved, and with the right tools and a bit of patience, you can solve any of them. Let's get started on this journey of mathematical discovery, and by the end, you'll be confidently solving equations like this one!

Step 1: Simplify Both Sides Using the Distributive Property

Alright team, the first move in our equation-solving playbook is to simplify both sides of the equation. Our equation is $5(x-1)+3x=7(x+1)$. Notice those parentheses? That's our cue to use the distributive property. This property basically says that if you have a number multiplied by a quantity inside parentheses, you multiply that number by each term inside the parentheses. So, on the left side, we have $5(x-1)$. Applying the distributive property here means we multiply 5 by 'x' and then multiply 5 by '-1'. This gives us $5x - 5$. Now, the left side of our equation becomes $5x - 5 + 3x$. We're not done simplifying the left side yet! See that we have two terms with 'x' in them? That's where we combine like terms. We have $5x$ and $3x$. When you add them together, $5x + 3x = 8x$. So, the entire simplified left side of our equation is now $8x - 5$. Easy peasy, right? Now, let's hop over to the right side of the equation: $7(x+1)$. Again, we use the distributive property. Multiply 7 by 'x', which gives us $7x$, and then multiply 7 by '+1', which gives us $+7$. So, the simplified right side of the equation is $7x + 7$. After this first step, our equation looks much cleaner: $8x - 5 = 7x + 7$. This is a huge step forward because it gets rid of the parentheses and makes the equation more manageable. Remember, the goal here is always to make the equation as simple as possible before we start moving terms around. Taking the time to correctly apply the distributive property and combine like terms now will save you a lot of headaches later on. It's like preparing your ingredients before you start cooking; you need everything neat and ready before you can create the final dish. So, pat yourselves on the back, you've just conquered the distributive property and combined like terms in a non-trivial equation! Now, we're perfectly set up for the next crucial stage of isolating our variable, 'x'. Don't forget the order of operations and the importance of treating both sides of the equation with equal respect. This initial simplification is fundamental to the entire process, setting a clean foundation for the algebraic manipulation that follows.

Step 2: Gather 'x' Terms on One Side

Okay, awesome job on simplifying, guys! Now that we have our cleaned-up equation, $8x - 5 = 7x + 7$, our next major goal is to get all the terms containing 'x' onto one side of the equation and all the constant terms (the numbers without 'x') onto the other side. This is where the magic of algebraic manipulation really comes into play. We want to isolate 'x', and you can't do that if it's chilling on both sides of the equals sign. Let's decide to move all the 'x' terms to the left side. Why the left? No particular reason, you could totally go for the right side too – it's your equation, after all! To move the $7x$ term from the right side to the left side, we need to do the opposite operation. Since $7x$ is currently positive on the right, we need to subtract $7x$ from both sides of the equation. This is super important – whatever you do to one side, you must do to the other to maintain balance. So, we'll subtract $7x$ from the left side and subtract $7x$ from the right side. On the left, we had $8x - 5$. Taking away $7x$ leaves us with $(8x - 7x) - 5$, which simplifies to $1x - 5$, or just $x - 5$. On the right side, we had $7x + 7$. Subtracting $7x$ from this gives us $(7x - 7x) + 7$, which simplifies to $0 + 7$, or just $7$. So, after subtracting $7x$ from both sides, our equation now looks like this: $x - 5 = 7$. See how much simpler that is? We've successfully gathered all our 'x' terms to one side. This step is crucial because it brings us one giant leap closer to having 'x' all by itself. It requires careful attention to signs – adding when you need to subtract, and subtracting when you need to add, to 'cancel out' terms. Remember, think of it as moving pieces on a chessboard; you're strategically positioning your variables and constants to achieve your objective. The key takeaway here is the inverse operation principle: to eliminate a term from one side, you perform its opposite operation on both sides. This ensures the equality remains true. You've now isolated the 'x' term, and the next step is incredibly straightforward, leading us directly to the final solution. Keep that momentum going, you're doing great!

Step 3: Isolate 'x' by Moving Constant Terms

We're on the home stretch, folks! Our equation is currently $x - 5 = 7$. We've managed to get all the 'x' terms to the left side, and now we just have a constant term, '-5', hanging out with our 'x'. Our ultimate goal is to have 'x' completely alone on one side. To do this, we need to get rid of that '-5'. Just like before, we use the principle of doing the opposite operation to both sides of the equation. Since we have '-5' on the left side, the opposite operation is to add 5. So, we add 5 to the left side and, you guessed it, we also add 5 to the right side to keep everything balanced. On the left side, we have $x - 5$. When we add 5 to it, we get $(x - 5) + 5$. The '-5' and '+5' cancel each other out, leaving us with just $x$. On the right side, we have the number 7. When we add 5 to it, we get $7 + 5$, which equals 12. So, after adding 5 to both sides, our equation becomes $x = 12$. Boom! We have found our solution! The value of 'x' that makes the original equation $5(x-1)+3x=7(x+1)$ true is 12. This step is often the quickest, as it involves dealing with just one constant term. It reinforces the idea that each step is designed to peel away layers, getting closer and closer to the core variable. The final isolation of 'x' is the culmination of all the previous efforts. It's like reaching the finish line after a challenging race; all the hard work pays off with that single, definitive answer. Remember to always perform the inverse operation – if a number is being added, subtract it from both sides; if it's being subtracted, add it to both sides. This consistent application of inverse operations is the key to successfully isolating any variable in an equation. You've conquered the equation, step by step, and arrived at the solution $x=12$. How satisfying is that?

Step 4: Verify Your Solution

Now, for the most satisfying part, guys: verifying our solution! It's always a good idea to double-check your work, especially in math. This step ensures that $x=12$ is indeed the correct answer to our original equation $5(x-1)+3x=7(x+1)$. To verify, we take our solution, $x=12$, and substitute it back into the original equation wherever we see an 'x'. Let's do the left side first: $5(x-1)+3x$. Substitute $x=12$ into this expression: $5(12-1)+3(12)$. First, calculate inside the parentheses: $12-1 = 11$. So, the expression becomes $5(11) + 3(12)$. Now, perform the multiplications: $5 imes 11 = 55$ and $3 imes 12 = 36$. Finally, add these results together: $55 + 36 = 91$. So, the left side of the original equation evaluates to 91 when $x=12$. Now, let's check the right side of the original equation: $7(x+1)$. Substitute $x=12$ into this expression: $7(12+1)$. First, calculate inside the parentheses: $12+1 = 13$. So, the expression becomes $7(13)$. Now, perform the multiplication: $7 imes 13 = 91$. Wowza! Both the left side and the right side of the equation evaluate to 91 when $x=12$. Since $91 = 91$, our solution is correct! This verification process is crucial. It doesn't just confirm your answer; it also helps reinforce your understanding of how variables function within equations. If we had gotten different numbers on each side, it would tell us that we made a mistake somewhere in our previous steps, and we'd need to go back and find it. This systematic checking prevents errors from snowballing and ensures accuracy. It's like proofreading your essay before submitting it – you catch any typos or grammatical errors. In mathematics, verification is your ultimate quality control. So, celebrate this moment – you didn't just solve an equation; you proved your solution! That's the power of algebraic thinking and rigorous checking. You've successfully navigated the entire process from a complex-looking equation to a verified, correct solution. Give yourselves a round of applause, you absolute math legends!

Conclusion: The Solution is x=12

So there you have it, guys! We've systematically dismantled the linear equation $5(x-1)+3x=7(x+1)$, step by painstaking step. We began by applying the distributive property to simplify both sides, transforming the equation into a more manageable form. Then, we cleverly gathered all the 'x' terms onto one side using inverse operations, paving the way for isolation. Following that, we isolated the 'x' variable by moving the constant term to the other side, revealing our potential solution. Finally, and perhaps most importantly, we verified our solution by plugging $x=12$ back into the original equation, confirming that both sides indeed yielded the same value. This meticulous process ensures not only that we arrive at the correct answer but also that our understanding of algebraic principles is solid. The final, verified solution to the equation $5(x-1)+3x=7(x+1)$ is unequivocally $x=12$. Remember these steps – simplify, gather variables, isolate the variable, and verify your answer. This approach is your golden ticket to conquering a wide array of linear equations you'll encounter. Keep practicing, keep questioning, and keep that mathematical curiosity alive. Every equation you solve builds your confidence and sharpens your problem-solving skills. This journey through a single equation illustrates the beauty and logic of mathematics. It’s a language, a tool, and a way of thinking that empowers you to understand and interact with the world around you. Don't shy away from these problems; embrace them as opportunities to learn and grow. Until next time, happy solving!