Solving Linear Equations A Step-by-Step Guide To 3x + 2(3x + 1) = 9x - 8

Hey guys! Ever stumbled upon a math problem that looks like it’s speaking a different language? Don't worry, we've all been there! Today, we’re going to break down a classic linear equation and solve it together. Our mission, should we choose to accept it (and we do!), is to tackle the equation 3x + 2(3x + 1) = 9x - 8. Sounds intimidating? Trust me, it's not as scary as it looks. We'll go through each step slowly, so you can follow along and become a master equation solver yourself. Linear equations are the building blocks of algebra, and mastering them opens up a whole new world of mathematical possibilities. Whether you're a student trying to ace your next math test or just someone who enjoys the thrill of problem-solving, understanding how to solve these equations is a valuable skill. So, grab your pencils, open your notebooks, and let's dive into the exciting world of algebra! Remember, the key to success in math is practice, practice, practice. The more you work through problems like this, the more comfortable and confident you'll become. So, let’s get started and unlock the mystery of this equation together!

Understanding Linear Equations

Before we jump into solving our specific equation, let’s take a moment to understand what linear equations are all about. Think of linear equations as mathematical sentences that describe a straight line when graphed. They involve variables (like our 'x') and constants, combined using basic arithmetic operations. The goal? To find the value of the variable that makes the equation true. These equations are called “linear” because the variable is only raised to the first power (no x², x³, etc.). They form straight lines when you graph them, hence the name. A linear equation typically looks like this: ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable we want to solve for. The variable 'x' represents an unknown quantity, and our job is to isolate it on one side of the equation to find its value. Understanding the structure of linear equations is crucial because it helps us apply the correct steps to solve them. For instance, we know that we need to perform inverse operations (like subtraction to undo addition) to isolate the variable. This foundational knowledge will make solving more complex equations much easier down the road. Linear equations are everywhere in real life, from calculating distances and speeds to figuring out budgets and investments. They’re a fundamental tool in science, engineering, economics, and many other fields. So, mastering them isn’t just about passing a math test; it’s about gaining a powerful problem-solving skill that you can use in countless situations. Let’s keep this in mind as we tackle our equation – we’re not just solving a math problem, we’re building a skill for life!

Step 1: Distribute

The first step in solving our equation, 3x + 2(3x + 1) = 9x - 8, is to simplify both sides as much as possible. Notice the term 2(3x + 1)? This is where the distributive property comes to our rescue! The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term inside a set of parentheses. Think of it like sharing: the 2 needs to be “shared” with both the 3x and the +1 inside the parentheses. So, we multiply 2 by 3x, which gives us 6x, and then we multiply 2 by +1, which gives us +2. This means we can rewrite 2(3x + 1) as 6x + 2. Now, our equation looks like this: 3x + 6x + 2 = 9x - 8. See how much simpler it's becoming already? Distributing is a crucial step because it eliminates the parentheses, making it easier to combine like terms and isolate the variable. Without distribution, we'd be stuck with a more complex expression that's harder to manage. This property isn't just useful for linear equations; it's a staple in algebra and is used extensively in more advanced math topics. It's like having a secret weapon that unlocks many mathematical problems! Remember, the distributive property is your friend. It helps you break down complex expressions into manageable parts. So, always look for opportunities to distribute when you're solving equations. It’s a powerful tool that will make your math life much easier. Now that we've distributed, we're one step closer to solving for 'x'. Let's move on to the next step and continue our journey to the solution!

Step 2: Combine Like Terms

Now that we've distributed and our equation looks like 3x + 6x + 2 = 9x - 8, it's time to combine like terms. What are like terms, you ask? They're terms that have the same variable raised to the same power. In our equation, 3x and 6x are like terms because they both have 'x' raised to the first power. The constant terms, like 2 and -8, are also like terms (they're just numbers!). To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). So, 3x + 6x becomes 9x. Now our equation looks even simpler: 9x + 2 = 9x - 8. This step is crucial because it reduces the number of terms in the equation, making it easier to isolate the variable. Think of it as tidying up – we're organizing the equation to make it more manageable. Combining like terms is a fundamental skill in algebra and is used in many different contexts. It's not just limited to solving equations; it's also used in simplifying expressions, working with polynomials, and more. Mastering this skill will make your algebraic journey much smoother. Remember, you can only combine terms that are truly alike. You can't combine 9x and 2 because one has the variable 'x' and the other is a constant. It’s like trying to add apples and oranges – they’re different things! So, always double-check that the terms you're combining have the same variable and power. Now that we've combined like terms, our equation is looking much cleaner and simpler. We're getting closer to isolating 'x' and finding our solution. Let's move on to the next step and see how we can further simplify the equation.

Step 3: Isolate the Variable

We've reached a crucial point in solving our equation 9x + 2 = 9x - 8: it's time to isolate the variable! This means we want to get all the terms with 'x' on one side of the equation and all the constant terms on the other side. To do this, we'll use inverse operations. Remember, whatever we do to one side of the equation, we must do to the other to keep it balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. Looking at our equation, we have 9x on both sides. To eliminate the 9x on the right side, we can subtract 9x from both sides. This gives us: 9x + 2 - 9x = 9x - 8 - 9x. Simplifying this, we get: 2 = -8. Wait a minute! Something interesting has happened. The 'x' terms have completely disappeared! This is a special case that tells us something important about the equation. When the variables cancel out, we're left with a statement that is either true or false. In this case, 2 = -8 is definitely a false statement. This means that there is no solution to the equation. No matter what value we plug in for 'x', the equation will never be true. This might seem like a disappointing outcome, but it's actually a valuable piece of information. It tells us that the equation is inconsistent, meaning there's no value of 'x' that satisfies it. Isolating the variable is a fundamental technique in solving equations, but it's also important to recognize when things don't go as expected. Sometimes, equations have one solution, sometimes they have infinite solutions, and sometimes, like in this case, they have no solution at all. Understanding these different possibilities is key to becoming a proficient equation solver. So, even though we didn't find a numerical value for 'x', we've still learned something important about the equation. Let's recap what we've done and solidify our understanding.

Step 4: Interpret the Result

So, we've gone through the steps: distributing, combining like terms, and attempting to isolate the variable in our equation 3x + 2(3x + 1) = 9x - 8. We ended up with the statement 2 = -8, which is clearly false. What does this mean? As we discussed, when the variables cancel out and we're left with a false statement, it means the equation has no solution. There's no value of 'x' that will make the equation true. Think of it like trying to find a number that is both even and odd at the same time – it's impossible! This type of equation is called an inconsistent equation. It's important to recognize these types of equations because they tell us that our initial problem might have some inherent contradictions. In real-world scenarios, this could indicate that our assumptions or models are flawed and need to be reevaluated. For example, if we were using this equation to model a physical system, the lack of a solution might suggest that the system is not behaving as we expected or that there's an error in our measurements. On the other hand, if we had ended up with a true statement (like 2 = 2), it would mean that the equation has infinite solutions. This type of equation is called an identity. Any value of 'x' would satisfy the equation. And, of course, if we had found a specific value for 'x', that would be the unique solution to the equation. Interpreting the result is just as important as the steps we take to solve the equation. It allows us to understand the bigger picture and draw meaningful conclusions. So, next time you're solving an equation, don't just stop at finding a value for the variable. Take a moment to think about what your result actually means. It might reveal something surprising! Now that we've interpreted our result, let's do a quick recap of the entire process to make sure we've got it all down.

Recap and Conclusion

Alright guys, let's take a step back and recap our journey through the equation 3x + 2(3x + 1) = 9x - 8. We started with a seemingly complex equation, but we broke it down into manageable steps. First, we used the distributive property to get rid of the parentheses, turning 2(3x + 1) into 6x + 2. This gave us the equation 3x + 6x + 2 = 9x - 8. Then, we combined like terms on the left side, simplifying 3x + 6x to 9x. Our equation now looked like 9x + 2 = 9x - 8. Next, we tried to isolate the variable 'x'. We subtracted 9x from both sides, which led to the surprising result: 2 = -8. This is where things got interesting! We realized that this is a false statement, meaning that our equation has no solution. It's an inconsistent equation. So, what did we learn? We learned that not all equations have a solution, and that's perfectly okay! Sometimes, the math is telling us that there's a contradiction in the problem itself. We also reinforced some key algebraic skills: the distributive property, combining like terms, and isolating the variable. These are fundamental tools that will help you tackle a wide range of mathematical problems. Remember, solving equations is like detective work. You're given a puzzle, and you need to use your skills and knowledge to find the hidden answer. Sometimes, the answer is a specific number. Sometimes, it's the realization that there is no answer. And sometimes, it's something else entirely! The key is to approach each problem with curiosity and a willingness to explore. So, keep practicing, keep asking questions, and keep exploring the wonderful world of mathematics. You've got this!