Solving The Equation: 2 - X - 4(-2x - 3) = -8x - 1

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Hey guys! Let's dive into solving this equation step by step. We're going to break it down so it's super clear how we get to the final answer. This kind of problem pops up a lot in algebra, and understanding the process is key. So, let's jump right in!

Understanding the Equation

Before we start crunching numbers, let's take a good look at the equation: 2 - x - 4(-2x - 3) = -8x - 1. At first glance, it might seem a bit intimidating with the parentheses and multiple terms. But don't worry, we'll tackle it piece by piece. Our main goal here is to isolate 'x' on one side of the equation. This means we need to simplify the equation by getting rid of the parentheses, combining like terms, and rearranging things until we have 'x' all by itself.

This equation involves several key algebraic concepts: the distributive property, combining like terms, and inverse operations. The distributive property is what we'll use to handle the parentheses. It basically says that a term multiplied by a group of terms inside parentheses is the same as multiplying that term by each term inside the parentheses individually. Combining like terms involves adding or subtracting terms that have the same variable and exponent. And inverse operations are the opposite operations we use to move terms around the equation – for example, adding to undo subtraction, or dividing to undo multiplication.

So, keep these concepts in mind as we go through the steps. We're going to use them all to solve this equation. Ready? Let's get started with the first step: dealing with those parentheses!

Step-by-Step Solution

1. Distribute the -4

The first thing we need to do is get rid of those parentheses. Remember the distributive property? We're going to multiply the -4 outside the parentheses by each term inside: (-2x) and (-3). So, -4 * -2x equals 8x, and -4 * -3 equals 12. Our equation now looks like this: 2 - x + 8x + 12 = -8x - 1.

Distributing correctly is crucial because it simplifies the equation and allows us to combine like terms later on. A common mistake is to only multiply -4 by the first term inside the parentheses, or to forget about the negative signs. So, always double-check your work at this stage to ensure accuracy. This step is all about expanding the expression and making it easier to work with. Once we've distributed, the equation becomes much clearer and we can move on to the next step: combining the like terms on each side.

2. Combine Like Terms

Now that we've gotten rid of the parentheses, let's simplify each side of the equation by combining like terms. On the left side, we have '-x' and '+8x', which combine to give us '7x'. We also have the constants '2' and '12', which add up to '14'. So, the left side of the equation simplifies to 7x + 14. Our equation now looks like this: 7x + 14 = -8x - 1.

Combining like terms helps to streamline the equation, making it less cluttered and easier to solve. This step is essential for organizing the equation and bringing similar terms together. By combining the 'x' terms and the constants, we're one step closer to isolating 'x'. It's like tidying up before we start the main work! Next, we're going to move all the 'x' terms to one side and the constants to the other. This will help us isolate 'x' and find its value.

3. Move x Terms to One Side

Our next goal is to get all the 'x' terms on one side of the equation. Let's move the '-8x' from the right side to the left side. To do this, we'll add '8x' to both sides of the equation. This is because adding is the inverse operation of subtracting. So, adding 8x to both sides cancels out the -8x on the right side. Our equation now looks like this: 7x + 8x + 14 = -1.

Adding '8x' to both sides keeps the equation balanced. Remember, whatever we do to one side, we must do to the other to maintain equality. Now, let's combine the 'x' terms on the left side. 7x + 8x equals 15x. So, our equation simplifies to: 15x + 14 = -1. We're making good progress! Now that we have all the 'x' terms on one side, we need to move the constants to the other side to further isolate 'x'.

4. Move Constants to the Other Side

Now, let's move the constant term, '14', from the left side to the right side. To do this, we'll subtract '14' from both sides of the equation. Subtracting is the inverse operation of adding, so this will cancel out the '14' on the left side. Our equation now looks like this: 15x = -1 - 14.

Subtracting '14' from both sides keeps the equation balanced, just like before. Now, let's simplify the right side. -1 - 14 equals -15. So, our equation now reads: 15x = -15. We're almost there! We've got 'x' on one side, multiplied by 15, and a constant on the other side. The final step is to isolate 'x' completely.

5. Isolate x

To get 'x' all by itself, we need to undo the multiplication. 'x' is currently being multiplied by '15', so we'll divide both sides of the equation by '15'. Dividing is the inverse operation of multiplying. Our equation now looks like this: x = -15 / 15.

Dividing both sides by '15' keeps the equation balanced, as always. Now, let's simplify the right side. -15 divided by 15 equals -1. So, we finally have our solution: x = -1.

Final Answer

So, the solution to the equation 2 - x - 4(-2x - 3) = -8x - 1 is x = -1. Awesome! We made it through all the steps and found our answer. It's a great feeling when you solve a tricky equation like this, right?

Verification

But hold on, before we celebrate too much, it's always a good idea to check our answer. How do we do that? Simple! We plug our solution, x = -1, back into the original equation and see if it holds true. This is called verification, and it's a crucial step to ensure we haven't made any mistakes along the way.

Let's substitute -1 for 'x' in the original equation: 2 - (-1) - 4(-2(-1) - 3) = -8(-1) - 1. Now, let's simplify step by step. First, we have 2 - (-1), which is the same as 2 + 1, giving us 3. Next, we need to deal with the parentheses: -2 * -1 equals 2, so we have 2 - 3 inside the parentheses, which equals -1. So, now we have 3 - 4(-1). Multiplying -4 by -1 gives us 4, so we have 3 + 4, which equals 7.

On the right side of the equation, we have -8 * -1, which equals 8, and then we subtract 1, giving us 7. So, both sides of the equation equal 7 when we substitute x = -1. This confirms that our solution is correct! Verification is your best friend in algebra – it gives you the confidence that you've nailed the problem.

Key Takeaways

Alright, guys, we've solved the equation and verified our answer. Let's recap the key steps we took:

  1. Distribute: We started by using the distributive property to eliminate the parentheses.
  2. Combine Like Terms: Next, we combined like terms on each side of the equation to simplify it.
  3. Move x Terms: We moved all the 'x' terms to one side of the equation by using inverse operations.
  4. Move Constants: Then, we moved all the constant terms to the other side, again using inverse operations.
  5. Isolate x: Finally, we isolated 'x' by dividing both sides by its coefficient, giving us our solution.
  6. Verification: We plugged the result into the original question to ensure it is the correct answer.

Remember, solving equations is like solving a puzzle. Each step brings you closer to the final answer. And don't forget the importance of checking your work – it can save you from making simple mistakes. Keep practicing, and you'll become a pro at solving algebraic equations in no time!