Solving The Equation 6(x-1) = 9(x+2) A Step-by-Step Guide
Hey guys! Let's dive into solving a classic algebraic equation today. We've got the equation 6(x-1) = 9(x+2), and our mission, should we choose to accept it, is to find the value of 'x' that makes this equation true. Don't worry, it's not as daunting as it looks! We'll break it down step-by-step, making sure everyone can follow along. Think of it like this: we're detectives, and 'x' is the mysterious suspect we need to identify. We'll use the clues (the numbers and operations in the equation) to track it down.
Step 1: Distribute the Numbers
The first thing we need to do is get rid of those parentheses. We do this by distributing the numbers outside the parentheses to the terms inside. Remember the distributive property? It basically means we multiply the number outside by each term inside. So, on the left side, we'll multiply 6 by both 'x' and '-1'. On the right side, we'll multiply 9 by both 'x' and '2'.
Let's break it down visually:
- 6 * x = 6x
- 6 * -1 = -6
- 9 * x = 9x
- 9 * 2 = 18
So, after distributing, our equation now looks like this: 6x - 6 = 9x + 18. See? We've already made progress! The equation is starting to look a lot less scary. It's like we've just opened the first door in our mathematical investigation. We're getting closer to our suspect, 'x'. At this stage, it is important to double check to avoid making any silly mistakes in the distribution process. A simple sign error can throw off the entire solution, so take a moment to ensure everything is correct.
Step 2: Gather the 'x' Terms
Now, we want 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 side. It's like we're sorting our evidence, putting all the 'x' clues in one pile and all the number clues in another. To do this, we'll use inverse operations. Inverse operations are just operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations.
Let's start by getting the 'x' terms on the left side. We have 6x on the left and 9x on the right. To get rid of the 9x on the right, we'll subtract 9x from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced. It's like a seesaw – if you add weight to one side, you have to add the same weight to the other side to keep it level.
So, we have:
6x - 6 - 9x = 9x + 18 - 9x
This simplifies to:
-3x - 6 = 18
Awesome! We've successfully moved the 'x' term to the left side. Now, let's move on to the next step. It's like we've narrowed down our suspect's location to a specific neighborhood. We're getting closer!
Step 3: Isolate the 'x' Term
Next up, we want to isolate the '-3x' term. This means we need to get rid of the '-6' that's hanging out on the left side. To do this, we'll use the inverse operation of subtraction, which is addition. We'll add 6 to both sides of the equation:
-3x - 6 + 6 = 18 + 6
This simplifies to:
-3x = 24
Great! We've isolated the '-3x' term. It's like we've found our suspect's house – we're almost there!
Step 4: Solve for 'x'
Finally, we need to solve for 'x'. Right now, we have '-3x', which means '-3 times x'. To get 'x' by itself, we need to undo the multiplication. The inverse operation of multiplication is division, so we'll divide both sides of the equation by -3:
-3x / -3 = 24 / -3
This simplifies to:
x = -8
Eureka! We've found our 'x'! The solution to the equation 6(x-1) = 9(x+2) is x = -8. It's like we've caught our suspect and solved the mystery. We can now confidently say that when x is -8, the equation holds true. To be absolutely sure, we can always plug our solution back into the original equation to verify.
Step 5: Verify the Solution (Optional but Recommended)
It's always a good idea to check your work, especially in math! To verify our solution, we'll plug x = -8 back into the original equation 6(x-1) = 9(x+2) and see if both sides are equal.
Let's substitute:
6(-8 - 1) = 9(-8 + 2)
Now, let's simplify:
6(-9) = 9(-6)
-54 = -54
It checks out! Both sides of the equation are equal, so we know that x = -8 is the correct solution. This step is crucial because it acts as a final confirmation, ensuring that we haven't made any errors along the way. It's like getting a second opinion from another detective to make sure we've got the right person. By verifying, we can be 100% confident in our answer.
Alternative Methods and Tips
While we've solved this equation using the distributive property and inverse operations, there are often alternative approaches you can take. For instance, in this case, you could have noticed that both 6 and 9 are divisible by 3. You could divide both sides of the original equation by 3 before distributing, which would result in smaller numbers and potentially simplify the calculations.
2(x - 1) = 3(x + 2)
Then you would proceed with the distribution and solving as before. This is a great tip for simplifying equations in general: always look for common factors that you can divide out to make the numbers smaller and more manageable. Another important tip is to stay organized. Write out each step clearly and neatly, so you can easily follow your work and catch any mistakes. It's also helpful to double-check your signs (positive and negative) throughout the process, as sign errors are a common source of mistakes in algebra.
Common Mistakes to Avoid
When solving equations like this, there are a few common mistakes that students often make. One mistake is forgetting to distribute to all the terms inside the parentheses. For example, in the first step, you need to multiply the 6 by both the 'x' and the '-1'. Make sure you don't leave out any terms. Another common mistake is making sign errors, especially when dealing with negative numbers. Pay close attention to the signs and double-check your work. A third mistake is not performing the same operation on both sides of the equation. Remember, to keep the equation balanced, whatever you do to one side, you must do to the other.
By being aware of these common mistakes, you can take steps to avoid them. Practice makes perfect, so the more equations you solve, the better you'll become at spotting potential errors and solving them correctly. It's like training your detective skills – the more cases you solve, the sharper your intuition becomes.
Real-World Applications
You might be wondering,