Solve 3(x-2)=22-x: How Many Solutions?

Hey math whizzes and equation explorers! Ever stare at an equation and wonder, "Just how many answers am I going to get here?" It's a totally valid question, guys, and today we're diving deep into the equation $3(x-2)=22-x$ to uncover its secrets. We're not just looking for the answer, but whether there's one, two, none, or even an infinite number of solutions. Get ready to flex those brain muscles because we're about to break down this algebraic puzzle step-by-step, making sure you feel super confident about identifying the number of solutions for any linear equation you throw our way. Let's get this party started!

Understanding the Equation $3(x-2)=22-x$

Alright, let's kick things off by looking closely at our equation: $3(x-2)=22-x$. This bad boy is what we call a linear equation in one variable, 'x'. The goal when we solve these types of equations is to isolate 'x' – to get it all by itself on one side of the equals sign. When we do that, we'll see exactly what value or values 'x' can take. The number of values 'x' can be is the number of solutions the equation has. So, how do we get 'x' alone? We use a series of algebraic steps, like balancing a scale. Whatever you do to one side of the equation, you must do to the other to keep things equal. This means we'll be using operations like addition, subtraction, multiplication, and division. Our main mission is to simplify both sides of the equation first, get all the 'x' terms together on one side, and then get all the constant numbers on the other. By the end of this process, we'll either find a specific value for 'x', discover that any value of 'x' works, or realize that no value of 'x' can satisfy the equation. It's like a detective story, and 'x' is the mystery we're trying to solve. The journey involves simplifying, gathering like terms, and using inverse operations, all while keeping that equation balanced. We'll be distributing that '3' on the left side, then moving terms across the equals sign. Each step is crucial in revealing the true nature of our solution set. So, buckle up, and let's start simplifying this equation to see where it leads us. Remember, the key is careful manipulation and a clear understanding of algebraic properties.

Step-by-Step Solution Process

Now, let's get our hands dirty and solve this equation, $3(x-2)=22-x$. First things first, we need to simplify the left side by distributing the '3' to both terms inside the parentheses. This gives us: $3*x - 3*2 = 22-x$, which simplifies to $3x - 6 = 22 - x$. See? Already looking a bit cleaner. Our next move is to get all the 'x' terms on one side and all the constant numbers on the other. Let's gather the 'x' terms first. We can add 'x' to both sides of the equation. This cancels out the '-x' on the right side and adds an 'x' to the left side. So, we have: $3x - 6 + x = 22 - x + x$. This simplifies to $4x - 6 = 22$. Now, we need to move the constant term '-6' from the left side to the right side. We do this by adding 6 to both sides: $4x - 6 + 6 = 22 + 6$. This results in $4x = 28$. We're almost there! The final step to isolate 'x' is to divide both sides by the coefficient of 'x', which is '4'. So, we get: $4x / 4 = 28 / 4$. And bam! We arrive at $x = 7$. So, we found a single, specific value for 'x' that makes the equation true. This process involved simplifying, distributing, and using inverse operations to isolate the variable. Each step was designed to maintain the equality of the equation, bringing us closer to the solution. It's like a carefully choreographed dance of numbers and operations, leading us to the unique answer. We successfully navigated through the algebraic landscape, and the result is a clear, individual value for 'x'. This straightforward approach is the standard for solving linear equations, ensuring clarity and accuracy in finding the solution.

Analyzing the Result: What Does $x=7$ Mean?

So, we've landed on $x=7$ as our solution. What does this actually mean in the grand scheme of things? It means that there is exactly one value for 'x' that makes the original equation $3(x-2)=22-x$ true. If you were to plug $x=7$ back into the original equation, both sides would perfectly balance out. Let's test it, shall we? On the left side, we have $3(7-2)$. First, calculate the part in the parentheses: $7-2 = 5$. Then, multiply by 3: $3 * 5 = 15$. So, the left side equals 15. Now, let's check the right side: $22-x$. Plugging in $x=7$, we get $22-7 = 15$. Lo and behold, both sides equal 15! This confirms that $x=7$ is indeed the correct and only solution. When solving a linear equation like this, finding a single numerical value for the variable indicates that the equation has one unique solution. This is the most common outcome for linear equations. It's like finding a specific key that perfectly fits a lock. The process of simplification and isolation led us directly to this single, definitive answer. The verification step is crucial because it solidifies our findings and assures us that our algebraic manipulations were accurate. It demonstrates that our solution is not just a guess, but a mathematically proven truth for the given equation. This one-to-one correspondence between the variable's value and the equation's truth is the hallmark of a unique solution. It’s a beautiful thing when everything just clicks into place, right?

Types of Solutions for Linear Equations

Now that we've cracked the case of $3(x-2)=22-x$, let's broaden our horizons and talk about the different types of solutions we might encounter when dealing with linear equations. While our equation gave us a nice, neat single solution, not all linear equations behave that way. There are generally three possibilities, guys:

1. One Unique Solution: This is what we just saw! When you solve the equation, you end up with a specific value for 'x' (like $x=7$). This means there's only one number that makes the equation true. Think of it as finding the exact right answer. This happens when the simplification process leads to a statement like $ax = b$, where 'a' is not zero. When you divide by 'a', you get a unique 'x'.

2. Infinitely Many Solutions: Sometimes, after you simplify and rearrange an equation, you might end up with something that looks like $0 = 0$. This is a true statement! What it means is that any real number you choose for 'x' will satisfy the equation. No matter what value you plug in for 'x', both sides of the equation will always be equal. This typically occurs when the variable terms on both sides cancel out completely, and the constant terms also end up being equal. For example, if you simplify an equation and get $5x + 3 = 5x + 3$, subtracting $5x$ from both sides leaves you with $3 = 3$. This identity means infinite solutions. It’s like having a universal key that opens every door!

3. No Solution: On the flip side, you might simplify an equation and arrive at a false statement, such as $0 = 5$ or $3 = 7$. This means there is no value of 'x' that can ever make the original equation true. No matter how hard you try, you'll never find an 'x' that satisfies it. This usually happens when the variable terms cancel out, but the constant terms are not equal. For instance, if simplifying leads to $2x + 4 = 2x + 1$, subtracting $2x$ from both sides leaves you with $4 = 1$, which is false. This contradiction tells you that the equation has no solution. It's like trying to fit a square peg in a round hole – it just won't work.

Understanding these three possibilities is super important because it helps you predict the outcome of solving an equation before you even start, or at least helps you interpret the result when you get something unexpected. It's all about recognizing those tell-tale identities ($0=0$) and contradictions ($0=5$) that pop up during the simplification process. These different scenarios help us categorize the nature of the relationship between the expressions on either side of the equals sign.

Identifying the Solution Type for $3(x-2)=22-x$

Let's bring it all back home to our specific equation: $3(x-2)=22-x$. We followed the steps, simplified, and isolated 'x'. What did we get? We got $x=7$. Remember how we talked about the three types of solutions? Since we arrived at a specific numerical value for 'x' that perfectly satisfies the equation, this means our equation falls into the first category: one unique solution. There isn't an infinite number of values for 'x' that work, nor is it impossible to find a value for 'x'. Instead, there is precisely one number, 7, that makes this equation a true statement. If we had ended up with something like $0=0$ during our simplification, we would have concluded infinitely many solutions. If we had encountered something like $5=10$, we would have declared no solution. But because we got $x=7$, we know for sure it's a single, distinct solution. This confirms that the equation is consistent and has a unique point of intersection if you were to graph both sides as separate lines. The algebraic process of simplification and isolation is designed to reveal this exact nature of the solution set. It’s the final destination of our algebraic journey, where the nature of 'x' is definitively revealed. It's a solid win for clarity and certainty in mathematics, proving that sometimes, there really is just one perfect answer.

Conclusion: One Solution Exists!

So, after all that algebraic wizardry, what's the verdict for the equation $3(x-2)=22-x$? We meticulously followed the steps, distributing, combining like terms, and isolating 'x'. Our journey led us to the definitive result: $x=7$. This outcome clearly indicates that there is one unique value for 'x' that satisfies the equation. It's not a situation of infinitely many solutions, nor is it a dead end with no solution. It’s a classic case of a linear equation with a single, specific answer. This means that when you plug $x=7$ back into the original equation, both the left side, $3(7-2)$, and the right side, $22-7$, will yield the same value (which we found to be 15). This confirms the accuracy of our solution and solidifies the fact that one solution exists. Keep practicing these steps, guys, and you'll become pros at spotting the number of solutions for any equation that comes your way. It’s all about the process, the simplification, and understanding what those final steps tell you about the nature of 'x'. Keep those math skills sharp!

Answer: C. one