Solving The Equation: $6(-4x-3) = -6(3x-3)$

Hey guys! Let's dive into solving this equation: $6(-4x-3) = -6(3x-3)$. It might look a little intimidating at first, but trust me, we'll break it down step-by-step, and it'll be a breeze. This is a fundamental concept in algebra, and understanding how to solve these types of equations is super important for your math journey. We'll be using the distributive property, combining like terms, and isolating the variable 'x' to find the solution. Ready? Let's get started!

Step-by-Step Solution: Unpacking the Equation

Alright, first things first, let's take a look at the equation again: $6(-4x-3) = -6(3x-3)$. The goal is to find the value of 'x' that makes this equation true. To do this, we need to simplify both sides of the equation. We'll start with the distributive property. This property tells us that we can multiply the number outside the parentheses by each term inside the parentheses. So, on the left side, we have $6 * -4x$ and $6 * -3$. On the right side, we have $-6 * 3x$ and $-6 * -3$. Let's do the multiplication:

• Left side: $6 * -4x = -24x$ and $6 * -3 = -18$. This gives us $-24x - 18$.
• Right side: $-6 * 3x = -18x$ and $-6 * -3 = 18$. This gives us $-18x + 18$.

Now, our equation looks like this: $-24x - 18 = -18x + 18$. See? It's already looking a bit cleaner. Remember, the distributive property is your friend when dealing with parentheses like this. It's all about making sure you multiply everything inside the parentheses by the number outside. Always double-check your signs – a small mistake there can lead to a big difference in your final answer. It is like the core of simplifying complex math problems. We use this all the time. Keep practicing, and you'll become a pro in no time. Next, let's move on to the next step, where we will move all the 'x' terms to one side of the equation and the constant terms to the other side. That will allow us to isolate 'x' and solve for its value.

Isolating the Variable

Okay, now that we've simplified, we need to get all the 'x' terms on one side and the constant numbers on the other side. This is called isolating the variable. It doesn't matter which side you choose, but let's move the 'x' terms to the left side and the constants to the right side. We can start by adding $18x$ to both sides of the equation. This cancels out the $-18x$ on the right side. So, we have:

$-24x - 18 + 18x = -18x + 18 + 18x$

Which simplifies to:

$-6x - 18 = 18$

Next, we need to get rid of the $-18$ on the left side. We do this by adding $18$ to both sides:

$-6x - 18 + 18 = 18 + 18$

This simplifies to:

$-6x = 36$

See how we're slowly but surely getting closer to solving for 'x'? We're almost there! Remember, the key is to perform the same operation on both sides of the equation to keep it balanced. It's like a seesaw – if you add or subtract something on one side, you have to do the same on the other side to keep it level. Combining like terms is the backbone of algebraic manipulation, as it reduces complexity. Always double-check your calculations to ensure everything balances. Now, one last step to solve for 'x'.

Solving for 'x'!

We're in the home stretch, guys! We've got $-6x = 36$. Now, to isolate 'x', we need to get rid of the $-6$ that's multiplying it. We do this by dividing both sides of the equation by $-6$:

$-6x / -6 = 36 / -6$

This simplifies to:

$x = -6$

And there you have it! The solution to the equation $6(-4x-3) = -6(3x-3)$ is $x = -6$. Congrats! You've successfully solved for 'x'. This is a fundamental skill in algebra, and you can apply this process to similar equations. It's all about practicing the steps: distribute, combine like terms, isolate the variable, and solve. Make sure to double-check your answers, and you will do great. If you encounter any problems, return to this step-by-step guide and review it. Now let's wrap this up, and let's go on to the next one!

Verification and Conclusion

Alright, we found that $x = -6$. But, how do we know if we are right? A great way to check is to substitute the value of 'x' back into the original equation and see if it holds true. Let's do that:

$6(-4(-6) - 3) = -6(3(-6) - 3)$

First, let's simplify inside the parentheses:

$6(24 - 3) = -6(-18 - 3)$

$6(21) = -6(-21)$

$126 = 126$

It checks out! Our solution, $x = -6$, is correct. Yay!

In conclusion, we've successfully solved the equation $6(-4x-3) = -6(3x-3)$. We used the distributive property to simplify, combined like terms, and isolated the variable 'x'. By following these steps and verifying our answer, we can be confident in our solution. Remember, practice is key, and with each equation you solve, you'll become more comfortable and proficient in algebra. Keep at it! This is a skill that will serve you well in math and beyond. Understanding these principles will make more complex math problems much easier to handle. So, keep practicing, and you'll find that these equations become easier with time.