Solving 2(x+3)=2x+6: How Many Solutions Exist?

Hey guys! Today, we're diving into a fun little math problem: figuring out the solution to the equation 2(x+3)=2x+6. This isn't just about getting an answer; it's about understanding what type of solution we're dealing with. Does it have one solution, no solutions, or an infinite number of solutions? Let's break it down step-by-step and make sure we understand the underlying concepts.

Understanding the Equation

Before we jump into solving, let's take a good look at the equation: 2(x+3)=2x+6. At first glance, it might seem like a standard linear equation. But, there's a little trick hidden inside. The key to unlocking this puzzle lies in simplifying and understanding what the equation is telling us. We have a variable, 'x,' and we need to figure out what value(s) of 'x' make the equation true. This is the fundamental concept of solving equations, and it's super important to grasp. In this particular case, we need to carefully examine the equation's structure to determine the nature of its solutions. We'll explore different methods to solve it and then analyze the results to categorize the solution type accurately. So, gear up, and let's dive deeper into the fascinating world of equation solving!

The Distributive Property

Okay, so the very first thing we need to tackle in our equation, 2(x+3)=2x+6, is the parentheses. Remember the distributive property? It's a crucial concept in algebra, and it's our best friend here. The distributive property basically says that if you have a number multiplied by a sum (or difference) inside parentheses, you multiply the number by each term inside the parentheses individually. It’s like sharing the love (or the multiplication, in this case!). So, for our equation, we're going to multiply the '2' outside the parentheses by both 'x' and '+3' inside. This step is absolutely critical because it helps us simplify the equation and reveal its true nature. Neglecting this step can lead to incorrect solutions or misinterpretations about the solution type. By correctly applying the distributive property, we transform the equation into a more manageable form, making it easier to analyze and solve. Now, let's move on to the exciting part: actually doing the distribution!

Applying the Distributive Property

Alright, let's put the distributive property into action! We have 2(x+3). So, we multiply 2 by x, which gives us 2x. Then, we multiply 2 by +3, which gives us +6. Easy peasy, right? This means 2(x+3) becomes 2x + 6. See how we've expanded the left side of the equation? This is a key transformation because now we can directly compare both sides of the equation. The distributive property is not just a mechanical rule; it's a tool that helps us uncover the structure of the equation. By applying it, we're essentially rearranging the terms in a way that makes the equation's properties more apparent. This step is often the gateway to understanding the equation's solution type, whether it's a unique solution, no solution, or infinitely many solutions. So, let's keep this in mind as we proceed with simplifying our equation!

Simplifying the Equation

Now that we've tackled the distributive property, our equation looks like this: 2x + 6 = 2x + 6. Take a good look at it. Notice anything interesting? The left side is exactly the same as the right side! This is a huge clue about the type of solution we're dealing with. When both sides of an equation are identical, it means that any value you plug in for 'x' will make the equation true. It's like saying 5 = 5, which is always true, no matter what. The simplification process has revealed a fundamental property of the equation. It's not just about finding a single value for 'x'; it's about recognizing that the equation represents an identity. This understanding is crucial for accurately classifying the solution type. So, let's explore what this means in terms of the number of solutions the equation has.

Identifying Identical Sides

Let's really focus on what it means to have identical sides in an equation. We've got 2x + 6 = 2x + 6. Imagine trying to solve this like a regular equation, where you're trying to isolate 'x'. You might subtract 2x from both sides, right? If you do that, you get 6 = 6. The 'x' terms are completely gone! This isn't a mistake; it's a powerful indication. When the variables disappear, and you're left with a true statement (like 6 = 6), it means the original equation is true for all possible values of 'x'. It’s like a universal truth within the realm of numbers. Recognizing this pattern is a key skill in algebra. It saves you time and helps you avoid the trap of trying to find a single solution where none exists. So, with this understanding, let's move on to the exciting part: determining the solution type!

Determining the Solution Type

We've simplified our equation down to 2x + 6 = 2x + 6, and we've seen that it's true no matter what 'x' is. So, what does this actually mean for our solution type? Well, it means we don't have just one solution, like x = 5, or x = -2. And it definitely doesn't mean we have no solution, because the equation is always true. Instead, we have a very special case: infinitely many solutions. Think about it – you could plug in any number for 'x', and the equation will still balance perfectly. This is because the equation is essentially a disguised identity. The two sides are fundamentally the same, just written in a slightly different form. Recognizing infinitely many solutions is a critical skill in algebra. It highlights the importance of simplifying and analyzing equations beyond just finding a single numerical answer.

What are Infinitely Many Solutions?

So, what does "infinitely many solutions" really mean? It sounds pretty grand, doesn't it? In the context of our equation, 2x + 6 = 2x + 6, it means there's no single value of 'x' that we can find that's the only answer. Instead, every single number you can think of will work. You could try x = 0, x = 1, x = -100, x = a million – they'll all make the equation true. This is because the equation is essentially saying something that's always true, regardless of the variable's value. It's a fundamental characteristic of equations that represent identities. The concept of infinitely many solutions is important because it broadens our understanding of what it means to solve an equation. It's not always about finding one specific answer; sometimes, it's about recognizing that the equation holds true across an entire spectrum of values. This understanding is crucial for tackling more complex mathematical problems in the future.

Final Answer and Explanation

Alright, guys, we've reached the finish line! We started with the equation 2(x+3) = 2x + 6, and after carefully applying the distributive property and simplifying, we discovered that both sides of the equation are identical. This crucial observation led us to the conclusion that the equation has infinitely many solutions. This means that any value we substitute for 'x' will satisfy the equation. There's no single, unique answer; the solution set encompasses all real numbers. This type of equation is called an identity, and it's a fascinating concept in algebra. Understanding identities is not just about solving this particular problem; it's about developing a deeper understanding of how equations work and the different types of solutions they can have. So, congratulations, guys, we've successfully navigated this math problem, and hopefully, you've gained a clearer understanding of infinitely many solutions!

Why Infinitely Many Solutions?

Let’s recap why infinitely many solutions is the correct answer. We began with 2(x+3) = 2x + 6. We used the distributive property to get 2x + 6 = 2x + 6. When the left and right sides of the equation are exactly the same, it signifies that no matter what value we assign to the variable 'x', the equation will always hold true. This is because both sides are essentially equivalent expressions. It's super important to remember this key concept: identical sides equal infinitely many solutions. This type of problem might seem tricky at first, but with practice and understanding of algebraic principles, you can easily identify and solve them. Remember, the goal isn't just to get the right answer, but to understand why it's the right answer. This deeper understanding will empower you to tackle more complex mathematical challenges with confidence!

So, the answer is C. Infinitely many solutions! We nailed it!