Infinite Solutions Unveiled Analyzing The Equation 3(x + 10) + 6 = 3(x + 12)
Hey there, math enthusiasts! Today, we're diving into a fascinating mathematical puzzle. We're going to explore the equation 3(x + 10) + 6 = 3(x + 12) and figure out exactly how many solutions it holds. Is it a single solution, none at all, a couple, or perhaps an infinite sea of possibilities? Get ready to sharpen your pencils (or fire up your favorite equation solver) as we break down this problem step by step. We'll not only find the answer but also uncover the underlying principles that make it tick. So, let's jump right in and decode this mathematical mystery together!
Unraveling the Equation: A Step-by-Step Journey
To determine the number of solutions for the equation 3(x + 10) + 6 = 3(x + 12), we need to embark on a step-by-step journey of simplification and analysis. This process will not only reveal the answer but also provide a deeper understanding of how equations behave. Think of it as detective work, where we follow the clues to uncover the truth hidden within the symbols.
Step 1: Distribute the Love (and the Numbers)
Our first move is to distribute the numbers outside the parentheses to the terms inside. This means multiplying the 3 by both the x and the 10 on each side of the equation. Remember, the distributive property is our friend here, ensuring we handle each term correctly. This step transforms our equation into something a bit more manageable and sets the stage for further simplification.
So, let's perform the distribution:
- Left side: 3 * x + 3 * 10 + 6 becomes 3x + 30 + 6
- Right side: 3 * x + 3 * 12 becomes 3x + 36
Now our equation looks like this: 3x + 30 + 6 = 3x + 36
Step 2: Combine Like Terms: Gathering the Troops
The next logical step is to combine like terms on each side of the equation. Like terms are those that have the same variable and exponent (in this case, just the constant terms). This is like gathering the troops before a battle, making sure we have all our forces aligned and ready to go. Combining like terms simplifies the equation further, bringing us closer to the solution.
On the left side, we have two constant terms: 30 and 6. Let's add them together:
30 + 6 = 36
Now our equation looks like this: 3x + 36 = 3x + 36
Step 3: The Moment of Truth: What Does It All Mean?
Now we've arrived at a crucial point. Our equation has been simplified to 3x + 36 = 3x + 36. Take a good look at this. What do you notice? The left side is exactly the same as the right side. This is a key observation that will unlock the mystery of the solutions.
What does it mean when both sides of an equation are identical? It means that no matter what value we substitute for x, the equation will always be true. Think about it: if you add 36 to 3 times any number, you'll get the same result as adding 36 to 3 times that same number. This is a fundamental property of equality.
Step 4: Decoding the Solution: Infinite Possibilities
Because the equation 3x + 36 = 3x + 36 is true for any value of x, we say that the equation has infinitely many solutions. This is because there's no single value of x that makes the equation true; rather, every single value works. It's like having a magic key that unlocks every door!
So, the answer to our initial question is that the equation 3(x + 10) + 6 = 3(x + 12) has infinitely many solutions. We've successfully navigated the steps of simplification and analysis to arrive at this conclusion.
Why Infinitely Many Solutions? Understanding Identities
Now that we've determined that the equation 3(x + 10) + 6 = 3(x + 12) has infinitely many solutions, let's delve a little deeper into the why. Understanding the underlying concept will not only solidify our understanding of this particular problem but also equip us to tackle similar situations in the future. We're not just looking for the answer; we're striving for true comprehension.
The Concept of Identities: Equations That Are Always True
The reason this equation has infinitely many solutions lies in the concept of identities. An identity is an equation that is true for all values of the variable. In other words, no matter what number you plug in for x, the equation will always hold. Our equation, 3(x + 10) + 6 = 3(x + 12), is a perfect example of an identity.
Think of it like a mirror: whatever is on one side is perfectly reflected on the other. There's no specific value of x that needs to be