Hey there, math enthusiasts! Today, we're diving into an exciting algebraic equation that might seem a bit tricky at first glance, but trust me, it's totally manageable. We're going to break down the equation 3(4x - 12) = 12(x - 3) step by step, so you'll not only understand how to solve it but also why each step is crucial. So grab your pencils, and let's get started!
Understanding the Basics of Equations
Before we jump into the specifics, let's quickly recap what an equation actually is. At its core, an equation is a mathematical statement that asserts the equality of two expressions. Think of it like a balancing scale: what's on one side must be equal to what's on the other side. Our mission? To find the value(s) of the variable (in this case, 'x') that makes this balance true. We aim to isolate 'x' on one side of the equation, which will give us our solution.
Now, solving equations involves using various operations, always ensuring we maintain the balance. Whatever we do to one side, we must do to the other. This includes addition, subtraction, multiplication, and division. The goal is to simplify the equation until we can clearly see what 'x' equals. Remember those order of operations rules (PEMDAS/BODMAS)? They will be our best friends here! So, with these basics in mind, let's tackle our equation:
Initial Assessment of the Equation
Looking at 3(4x - 12) = 12(x - 3), the first thing that pops out is the parentheses. They're a clear sign that we'll need to use the distributive property. This property is a fundamental concept in algebra, and it's the key to unlocking the equation. The distributive property states that a(b + c) = ab + ac. In simpler terms, it means we multiply the term outside the parentheses by each term inside the parentheses. This will help us eliminate the parentheses and make the equation easier to work with. We also notice that there are terms with 'x' on both sides of the equation. This is pretty common, and our strategy will be to gather all the 'x' terms on one side and the constants on the other. This makes it simpler to isolate the variable and solve for 'x'. Keep this overview in mind as we move forward, and you'll see how each step contributes to the final solution. Alright, let's get into the nitty-gritty of solving this equation. The key to unraveling this equation lies in simplifying each side, and that's exactly what we're going to do. By applying fundamental algebraic principles, we will transform this seemingly complex equation into a much simpler form.
Step-by-Step Solution
1. Apply the Distributive Property
The first order of business is to eliminate those parentheses. This is where the distributive property comes in handy. Let’s apply it to both sides of the equation:
- Left Side: 3 * (4x - 12) = (3 * 4x) - (3 * 12) = 12x - 36
- Right Side: 12 * (x - 3) = (12 * x) - (12 * 3) = 12x - 36
So, after applying the distributive property, our equation now looks like this: 12x - 36 = 12x - 36. See how much cleaner it looks already? This step is crucial because it breaks down the equation into simpler terms, making the next steps more straightforward. By distributing the numbers outside the parentheses, we've essentially opened up the equation, allowing us to see the relationship between the terms more clearly. Now that we've got rid of the parentheses, we're one step closer to isolating 'x' and finding our solution.
2. Simplify the Equation
Now that we've applied the distributive property, we have a new equation: 12x - 36 = 12x - 36. Take a good look at this equation. Notice anything interesting? You might see that both sides are exactly the same! This is a huge clue. When both sides of an equation are identical, it means that the equation is an identity. An identity is an equation that is true for any value of the variable. Think about it: no matter what number we substitute for 'x', the left side will always equal the right side. This is a unique situation, and it tells us something important about the solution to the equation. It's not just one specific value of 'x' that works; it's any value! But let's continue with our typical solving process just to illustrate this point further. We'll try to isolate 'x' as we normally would, and you'll see how the equation leads us to this conclusion.
3. Isolate the Variable (x)
To isolate 'x', we want to get all the 'x' terms on one side of the equation and the constants on the other. Let's subtract 12x from both sides: (12x - 36) - 12x = (12x - 36) - 12x. This simplifies to -36 = -36. Notice that the 'x' terms have completely disappeared! This is another indicator that we're dealing with an identity. When the variable vanishes during the solving process, and we're left with a true statement (like -36 = -36), it confirms that any value of 'x' will satisfy the original equation. Now, if we were to try and continue solving for 'x', we wouldn't get a specific value. We've reached a point where the equation is true regardless of 'x'. This is a crucial concept in algebra, and it's important to recognize these situations when they arise. So, what does this tell us about the solution to our equation? Let's discuss that in the next section.
Interpreting the Solution
So, we've arrived at the equation -36 = -36. As we discussed, this is a true statement, but there's no 'x' left to solve for! What does this mean in the context of our original problem? Well, it tells us that the equation 3(4x - 12) = 12(x - 3) is an identity. In other words, it's true for all values of 'x'. You could plug in any number for 'x' – whether it's a positive number, a negative number, a fraction, or even zero – and the equation will always hold true. This might seem a bit strange at first, especially if you're used to equations having a single solution or no solution at all. But identities are an important part of algebra, and recognizing them is a valuable skill. They show us that sometimes, equations aren't about finding one specific answer; they're about understanding relationships that are universally true. When you encounter an identity, it's like discovering a fundamental truth about the mathematical expressions involved. So, in this case, the solution isn't just one number; it's the entire set of real numbers. This is often represented as
