Easy Algebra: Solve $5(3+2x)=9(x-4)$ Now!

Alright guys, let's dive into some algebra today with a problem that might look a little tricky at first glance, but trust me, it's totally manageable. We're going to tackle the equation: $5(3+2x)=9(x-4)$. This kind of problem is super common in math classes, and mastering it will give you a solid foundation for more complex equations down the line. So, grab your notebooks, maybe a snack, and let's break this down step-by-step. We want to find the value of 'x' that makes this equation true. Think of it like a balancing scale; whatever we do to one side, we have to do to the other to keep it balanced. Our main goal here is to isolate 'x' on one side of the equation. We'll be using the distributive property, combining like terms, and a few other algebraic maneuvers. Don't get intimidated by the numbers and variables; we'll go through each part slowly and make sure it all makes sense. By the end of this, you'll not only have solved this specific equation but also gained confidence in your algebraic problem-solving skills. So, let's get started on unlocking the mystery of 'x' in $5(3+2x)=9(x-4)$.

Understanding the Equation: $5(3+2x)=9(x-4)$

So, the first thing we need to do when we look at an equation like $5(3+2x)=9(x-4)$ is to understand what's going on. We've got parentheses, numbers outside them, and variables inside. This setup means we need to use the distributive property. Remember that? It's like saying the number outside the parentheses gets to 'distribute' itself – meaning multiply – to each term inside. On the left side, we have '5' multiplying both '3' and '2x'. On the right side, '9' is multiplying both 'x' and '-4'. This is the crucial first step because it gets rid of the parentheses and makes the equation look a lot simpler to work with. After we apply the distributive property, we'll have a new, equivalent equation without any parentheses. This is a fundamental technique in algebra, and it's essential for simplifying expressions and solving equations. It allows us to expand terms and often reveal the underlying structure of the problem. Think of it as unfolding a gift box; you're revealing what's inside by opening it up. The numbers outside the brackets are like the wrapping paper, and the terms inside are the presents. You can't really get to the presents without dealing with the wrapping. So, let's prepare to unwrap this equation using the distributive property, which is going to be our key to moving forward. This initial step is all about transforming the equation into a more manageable form, paving the way for us to isolate the variable 'x'. We're essentially rewriting the equation in a way that's easier to manipulate, bringing us closer to finding that elusive value of 'x'.

Step 1: Applying the Distributive Property

Okay, guys, let's get our hands dirty with the first step: applying the distributive property to $5(3+2x)=9(x-4)$. On the left side, we take the '5' and multiply it by the '3', and then we take the '5' and multiply it by the '2x'. So, $5 * 3$ gives us $15$. And $5 * 2x$ gives us $10x$. Putting that together, the left side of our equation becomes $15 + 10x$. Now, let's move to the right side. We take the '9' and multiply it by the 'x', which gives us $9x$. Then, we take the '9' and multiply it by the '-4'. Remember, a positive number multiplied by a negative number results in a negative number, so $9 * -4$ is $-36$. So, the right side of our equation becomes $9x - 36$. After applying the distributive property to both sides, our original equation $5(3+2x)=9(x-4)$ has transformed into a new, equivalent equation: $15 + 10x = 9x - 36$. This step is huge because it removes the parentheses, which often makes equations look more intimidating than they are. Now, we have an equation with terms that are easier to combine and rearrange. This is where the real work of isolating 'x' begins. By distributing, we've essentially laid out all the components of the equation in a more straightforward manner, setting us up perfectly for the next stages of simplification and solving. It's like clearing the table before you start cooking – you have all your ingredients prepped and ready. The distributive property is a cornerstone of algebraic manipulation, and mastering it is key to unlocking proficiency in solving a wide range of mathematical problems. This transformation is crucial, and it's important to be meticulous with your calculations here to ensure accuracy in the subsequent steps.

Step 2: Gathering Variable Terms

Now that we've got our equation looking like $15 + 10x = 9x - 36$, our next mission, should we choose to accept it, is to get all the terms with 'x' on one side of the equation and all the constant terms (the numbers without 'x') on the other. This is how we start to isolate 'x'. A good strategy is to move the smaller 'x' term to the side with the larger 'x' term to keep our numbers positive, if possible. In this case, we have $10x$ on the left and $9x$ on the right. Since $10x$ is bigger than $9x$, let's move the $9x$ to the left side. To do this, we need to perform the opposite operation. Since $9x$ is being added on the right side (even though it's written as $9x - 36$, the $9x$ itself is positive), we'll subtract $9x$ from both sides of the equation to keep it balanced. So, on the left side, we'll have $15 + 10x - 9x$. Combining the 'x' terms, $10x - 9x$ gives us just $1x$, or simply $x$. So the left side becomes $15 + x$. On the right side, we'll have $9x - 36 - 9x$. The $9x$ and $-9x$ cancel each other out, leaving us with just $-36$. So, our equation now looks like $15 + x = -36$. See how we're getting closer? We've successfully moved all the 'x' terms to one side. This process of moving terms across the equals sign by using inverse operations is fundamental to solving equations. It's like sorting items into different boxes; you want all the 'x' items in one box and all the number items in another. By subtracting $9x$ from both sides, we've ensured that the equality remains true while systematically rearranging the equation to bring us nearer to our final answer. This step requires careful attention to signs – adding when you need to subtract, and vice versa – to maintain the balance of the equation.

Step 3: Isolating the Variable

We're on the home stretch, people! Our equation is currently $15 + x = -36$. The goal is to get 'x' all by itself. Right now, we have '15' added to 'x'. To isolate 'x', we need to undo that addition. The opposite of adding 15 is subtracting 15. So, just like before, we must subtract 15 from both sides of the equation to maintain the balance. On the left side, we have $15 + x - 15$. The $+15$ and $-15$ cancel each other out, leaving us with just $x$. On the right side, we have $-36 - 15$. When you subtract a positive number from a negative number, you're essentially moving further down the number line into more negative territory. So, $-36 - 15$ equals $-51$. Therefore, our equation becomes $x = -51$. And there you have it! We've successfully isolated 'x' and found its value. This final step of isolating the variable is all about performing the inverse operation of whatever is being done to 'x'. Whether it's addition, subtraction, multiplication, or division, we use the opposite operation to get 'x' alone. It's a satisfying moment when you reach this point, as it means you've navigated all the complexities of the equation and arrived at the solution. This final move is critical, and just like before, consistency across both sides of the equals sign is paramount to ensuring the accuracy of our final answer. The ability to perform these inverse operations confidently is a hallmark of strong algebraic skills.

Step 4: Checking Your Solution

So, we found that $x = -51$. But are we sure it's correct? In math, especially in algebra, it's always a brilliant idea to check your answer. This is like double-checking your work before handing in a big test. To check our solution for $5(3+2x)=9(x-4)$, we substitute $-51$ back into the original equation wherever we see 'x'. Let's do the left side first: $5(3 + 2 * (-51))$. Inside the parentheses, $2 * (-51)$ is $-102$. So, we have $5(3 - 102)$. $3 - 102$ is $-99$. Now, $5 * (-99)$ is $-495$. So, the left side of the equation evaluates to $-495$. Now, let's check the right side with $x = -51$: $9((-51) - 4)$. Inside the parentheses, $-51 - 4$ is $-55$. So, we have $9 * (-55)$. $9 * (-55)$ is also $-495$. Since the left side ($-495$) equals the right side ($-495$), our solution $x = -51$ is absolutely correct! This checking step is super important. It not only confirms your answer but also helps you catch any mistakes you might have made during the solving process. If the two sides didn't match, it would mean there was an error somewhere, and we'd have to go back and review our steps. This process of substitution and verification reinforces your understanding of how equations work and builds confidence in your abilities. Always make it a habit to check your answers, guys; it's a pro move in the world of math! This verification process is the ultimate confirmation that our algebraic journey has led us to the right destination, solidifying the correctness of our findings.