Solving 3(x-2) = 2(x-3): A Step-by-Step Guide
Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a tangled mess of numbers and parentheses? Don't worry, we've all been there. Today, we're going to break down a classic equation, 3(x-2) = 2(x-3), and walk through each step to find the solution. But more than just finding 'x,' we'll focus on why each step works. We're not just memorizing; we're understanding the logic behind the math. So, buckle up, grab your calculators (or not, you might not even need them!), and let's dive into the wonderful world of equation solving!
The Initial Equation: Setting the Stage
Okay, so we're starting with the equation: 3(x - 2) = 2(x - 3). This might look a bit intimidating at first glance, but trust me, it's totally manageable. The key here is to remember that an equation is like a balanced scale. Whatever we do to one side, we must do to the other to keep things equal. Think of it as mathematical karma – what goes around comes around, but in a good, solution-finding way! Now, before we start moving things around, we need to simplify each side individually. That's where our first big step comes in: the Distributive Property.
Step 1: Unleashing the Distributive Property
Ah, the Distributive Property, a true hero in the world of algebra! This property basically tells us how to handle those pesky parentheses. When we see a number multiplied by something in parentheses, like in our equation, we need to "distribute" that number to each term inside. In simpler terms, we multiply the number outside the parentheses by everything inside. So, for the left side of our equation, 3(x - 2), we multiply the 3 by both 'x' and '-2'. This gives us 3 * x = 3x and 3 * -2 = -6. Combining these, the left side simplifies to 3x - 6. We do the same thing on the right side with 2(x - 3). Multiplying 2 by 'x' gives us 2x, and multiplying 2 by '-3' gives us -6. So, the right side simplifies to 2x - 6. Now, our equation looks much cleaner: 3x - 6 = 2x - 6. See? Not so scary after all! We've used the Distributive Property to expand both sides, getting rid of those parentheses and making our equation easier to work with. This step is crucial because it allows us to isolate the 'x' terms and the constant terms, bringing us closer to our final solution. Without distributing, we'd be stuck with those parentheses, and it would be much harder to move things around and solve for 'x'. So, always remember the Distributive Property – it's your friend in the equation-solving game!
Step 2: Leveraging the Power of Addition and Subtraction Properties of Equality
Now that we've tamed those parentheses using the Distributive Property, it's time to gather like terms. We want all our 'x' terms on one side of the equation and all our constant terms (the numbers without 'x') on the other side. This is where the Addition and Subtraction Properties of Equality come into play. These properties are like the golden rules of equation solving. They state that we can add or subtract the same value from both sides of an equation without changing its balance. Remember that balanced scale? We're keeping it level! Looking at our equation, 3x - 6 = 2x - 6, we have 'x' terms on both sides. To get them together, let's subtract 2x from both sides. This cancels out the 2x on the right side and leaves us with just the constant term. Subtracting 2x from the left side, 3x - 2x simplifies to x. Our equation now looks like x - 6 = -6. Awesome! We've managed to get all the 'x' terms on one side. Now, let's deal with the constant terms. We have a -6 on the left side that we want to get rid of. To do this, we use the Addition Property of Equality. We add 6 to both sides of the equation. On the left side, -6 + 6 cancels out, leaving us with just 'x'. On the right side, -6 + 6 also equals zero. So, our equation magically transforms into x = 0. Ta-da! We've solved for 'x'! By using the Addition and Subtraction Properties of Equality, we've carefully moved terms around, isolating 'x' and revealing its value. These properties are essential tools in any equation solver's arsenal. They allow us to manipulate equations in a controlled way, always maintaining balance and leading us closer to the solution.
Step 3: Multiply to Simplify (Not Applicable in This Case, but Important to Know!)
In our specific equation, 3(x-2) = 2(x-3), we didn't actually need to use multiplication as a standalone step to isolate 'x' after applying the distributive property and addition/subtraction properties. However, it's super important to understand why multiplication (and its inverse, division) can be a crucial step in solving other equations. Think of it this way: sometimes, 'x' might be trapped in a fraction or multiplied by a coefficient that we need to get rid of. That's where multiplication and division come to the rescue. Let's imagine a slightly different scenario. Suppose we had an equation like 2x = 4. Here, 'x' is being multiplied by 2. To isolate 'x', we would perform the inverse operation: division. We would divide both sides of the equation by 2. This would cancel out the 2 on the left side, leaving us with just 'x', and on the right side, 4 divided by 2 equals 2. So, we'd get x = 2. Similarly, if we had an equation like x/3 = 5 (where 'x' is being divided by 3), we would use multiplication to isolate 'x'. We would multiply both sides of the equation by 3. This would cancel out the division by 3 on the left side, leaving us with 'x', and on the right side, 5 multiplied by 3 equals 15. So, we'd get x = 15. The key takeaway here is that multiplication and division are powerful tools for undoing operations and isolating 'x'. They're the inverse operations of each other, and they work together to help us solve a wide range of equations. While we didn't need them in this particular example, it's crucial to have them in your mathematical toolkit for future equation-solving adventures!
The Grand Finale: x = 0! A Victory for Algebra!
After carefully navigating through the Distributive Property and the Addition and Subtraction Properties of Equality, we've arrived at our destination: x = 0! Give yourself a pat on the back, guys – you've successfully solved the equation. But let's not just stop at the answer. Let's take a moment to appreciate the journey. We started with an equation that looked a bit complex, with parentheses and 'x' terms scattered on both sides. But by systematically applying the rules of algebra, we were able to simplify it step by step, until we isolated 'x' and revealed its value. This process highlights the beauty and power of mathematics. It's not just about memorizing formulas; it's about understanding the underlying logic and using it to solve problems. Each step we took was based on a fundamental principle, whether it was the Distributive Property or the Properties of Equality. And by following these principles, we were able to transform a seemingly complicated equation into a simple and elegant solution. So, the next time you encounter an equation that looks challenging, remember this journey. Remember the steps we took, the properties we used, and the satisfaction of reaching the final answer. And remember that with a little bit of knowledge and a lot of logical thinking, you can conquer any mathematical challenge that comes your way! Congratulations on solving this equation. You're well on your way to becoming an equation-solving pro!
