Solve For X: -6x + 3(x-4) = -9

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the world of algebra to tackle a problem that might look a little intimidating at first glance, but trust me, it's totally manageable. We're going to solve for $x$ in the equation $-6x+3(x-4)=-9$. This is a fundamental skill in mathematics, and understanding how to isolate a variable is like unlocking a secret code in a lot of different fields, not just math class. Whether you're a seasoned math whiz or just starting to explore the wonders of equations, this breakdown is for you. We'll go step-by-step, making sure every move is clear and easy to follow. So, grab your calculators, maybe a fresh cup of coffee, and let's get this done!

Our mission, should we choose to accept it, is to find the value of $x$ that makes the equation $-6x+3(x-4)=-9$ true. Think of it like a balancing act. Whatever we do to one side of the equation, we must do to the other side to keep things equal. The ultimate goal is to get $x$ all by itself on one side of the equals sign. To do this, we'll use a combination of algebraic operations: distribution, combining like terms, addition, and subtraction. Don't worry if some of these terms sound like jargon; we'll explain everything as we go. The first hurdle we need to clear is that pesky set of parentheses. That $3$ outside the $(x-4)$ means we need to multiply it by both the $x$ and the $-4$ inside. This is called the distributive property, and it's a crucial first step in simplifying expressions like this. Once we've distributed, we'll have a new version of our equation, and then we can start simplifying further by gathering all the terms involving $x$ together and all the constant terms together. It's all about making the equation cleaner and easier to work with. So, let's roll up our sleeves and start unpacking this equation, one step at a time.

Step 1: Distribute to Simplify the Equation

Alright, first things first, let's tackle that distributive property. In our equation, $-6x+3(x-4)=-9$, we have $3$ multiplied by the expression inside the parentheses, $(x-4)$. This means we need to distribute that $3$ to both the $x$ and the $-4$. So, $3$ times $x$ is $3x$, and $3$ times $-4$ is $-12$. Now, we can rewrite the equation without the parentheses. Our original equation was $-6x+3(x-4)=-9$. After distributing, it becomes $-6x + 3x - 12 = -9$. See? It's already looking a bit more streamlined. We've eliminated the parentheses, which is often the trickiest part for beginners. Remember, the distributive property is your best friend when you see a number right next to a set of parentheses. It's basically saying, 'Hey, this number outside needs to say hello to everyone inside!' Just make sure you multiply by every term within the parentheses and pay close attention to the signs. If the number outside is negative, multiplying by it will flip the signs of the terms inside. In this case, our $3$ was positive, so the signs of $x$ and $-4$ remained the same when multiplied. This simplification is key because it sets us up for the next phase: combining like terms. Without this step, trying to combine terms would be much more confusing. So, celebrate this small victory – we've made the equation simpler and closer to revealing the value of $x$!

Step 2: Combine Like Terms

Now that we've distributed and simplified, our equation is $-6x + 3x - 12 = -9$. The next logical move is to combine what we call 'like terms'. In algebra, like terms are terms that have the same variable raised to the same power. In our equation, $-6x$ and $3x$ are like terms because they both have the variable $x$ to the power of 1 (which we usually don't write). The numbers $-12$ and $-9$ are also like terms because they are both constants (numbers without any variables). Our goal here is to consolidate these terms to make the equation even more compact. Let's focus on the $x$ terms first: we have $-6x + 3x$. When we combine these, we're essentially adding their coefficients (the numbers in front of the $x$). So, $-6 + 3$ equals $-3$. Therefore, $-6x + 3x$ simplifies to $-3x$. Now, let's look at the constants. We have $-12$ on the left side and $-9$ on the right side. For now, we just identify them. The equation now looks like $-3x - 12 = -9$. Combining like terms is super important because it reduces the number of terms we have to deal with. It's like decluttering your workspace; the cleaner it is, the easier it is to focus on the task at hand. Remember, when combining terms with different signs, you subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. In our case, $6$ is larger than $3$, so we keep the negative sign. This step brings us one giant leap closer to isolating $x$!

Step 3: Isolate the Variable Term

We're on a roll, guys! Our equation has been simplified to $-3x - 12 = -9$. The next major step is to isolate the term that contains our variable, which is $-3x$. To do this, we need to get rid of that $-12$ on the left side of the equation. Remember our balancing act? Whatever we do to one side, we must do to the other. Since we have $-12$ on the left, we need to perform the opposite operation to cancel it out. The opposite of subtracting $12$ is adding $12$. So, we will add $12$ to both sides of the equation. Let's see how that plays out:

$-3x - 12 + 12 = -9 + 12$

On the left side, $-12 + 12$ cancels out to $0$, leaving us with just $-3x$. On the right side, $-9 + 12$ equals $3$. So, our equation now becomes $-3x = 3$. Isn't that neat? We've successfully moved the constant term away from the variable term, leaving $-3x$ all by itself on one side. This is a critical stage because it means we're just one step away from finding the exact value of $x$. Always remember that to isolate a term, you use the inverse operation. If a number is being added, subtract it. If it's being subtracted, add it. This principle applies to multiplication and division too, which we'll see in the next step. Keeping the equation balanced is paramount, so adding $12$ to both sides ensures we maintain that equality.

Step 4: Solve for x

We've reached the final frontier, the moment of truth! Our equation is currently $-3x = 3$. This means $-3$ is being multiplied by $x$. To get $x$ all by its lonesome, we need to undo that multiplication. The inverse operation of multiplication is division. So, we need to divide both sides of the equation by the coefficient of $x$, which is $-3$. Let's do the math:

rac{-3x}{-3} = rac{3}{-3}

On the left side, the $-3$ in the numerator and the $-3$ in the denominator cancel each other out, leaving us with just $x$. On the right side, we have $3$ divided by $-3$. A positive number divided by a negative number results in a negative number. So, $3 ext{ divided by } -3$ is $-1$. Therefore, our solution is $x = -1$.

We've officially solved for $x$! This means that if you substitute $-1$ back into the original equation $-6x+3(x-4)=-9$, the equation will hold true. Let's quickly check this: $-6(-1) + 3(-1 - 4) = 6 + 3(-5) = 6 - 15 = -9$. It works!

Conclusion: The Power of Algebraic Steps

So there you have it, guys! We successfully solved the equation $-6x+3(x-4)=-9$ and found that $x = -1$. We walked through each step methodically: distributing to remove parentheses, combining like terms to simplify, isolating the variable term by using inverse operations, and finally, dividing to solve for $x$. This process might seem straightforward for this particular problem, but the principles we used are universal for solving a vast array of algebraic equations. Understanding how to manipulate equations, keep them balanced, and isolate variables is an incredibly powerful skill. It's not just for math tests; it's a fundamental tool for problem-solving in science, engineering, finance, and even everyday decision-making. Keep practicing, and don't be afraid to tackle more complex equations. The more you practice, the more comfortable and confident you'll become. Keep your eyes on Plastik Magazine for more math breakdowns and fascinating discussions. Until next time, happy solving!