Solving 3x + 9 = 15x + 23: Step-by-Step

Hey guys, let's dive into solving algebraic equations! Today, we're tackling a pretty common type: 3x + 9 = 15x + 23. Don't let the numbers and letters scare you; we'll break it down step-by-step. Our main goal here is to find the value of 'x' that makes this equation true. Think of it like a puzzle where 'x' is the missing piece. We'll use a few fundamental rules of algebra to isolate 'x' on one side of the equation. It’s all about keeping things balanced, like a scale. Whatever you do to one side, you must do to the other to maintain that balance. This principle is super important and will be our guiding light as we move through the problem.

To start, we want to get all the 'x' terms together and all the constant numbers together. It doesn't matter which side you choose for 'x', but it's often easier to move the smaller 'x' term to the side with the larger 'x' term to keep our numbers positive. In our equation, 3x + 9 = 15x + 23, the 'x' terms are 3x and 15x. Since 15x is larger than 3x, let's move the 3x. To do this, we subtract 3x from both sides of the equation. This is our first balancing act! So, we have (3x + 9) - 3x = (15x + 23) - 3x. On the left side, the 3x and -3x cancel each other out, leaving us with just 9. On the right side, 15x minus 3x gives us 12x. So now, our equation looks much simpler: 9 = 12x + 23. See? We're already making progress! This is the beauty of algebraic manipulation; with each step, the equation becomes more manageable. Remember, the key is to perform the inverse operation. Since we're adding 3x to one side, we subtract 3x to undo it. This is a core concept in solving for any variable.

Now that we've grouped our 'x' terms, let's focus on the constant numbers. Our current equation is 9 = 12x + 23. We want to get the '12x' term by itself. To do that, we need to move the '+ 23' away from it. The opposite of adding 23 is subtracting 23. So, we subtract 23 from both sides of the equation. Again, it's all about balance! So, we have 9 - 23 = (12x + 23) - 23. On the left side, 9 minus 23 gives us -14. On the right side, the +23 and -23 cancel out, leaving us with just 12x. Our equation is now even closer to the solution: -14 = 12x. This is fantastic! We've successfully isolated the term with 'x'. This stage often feels like the biggest hurdle has been cleared, and you're right there. The structure of the equation has been simplified to a point where the final step is straightforward. Each step taken is a direct application of the additive inverse property, ensuring that the equality holds true throughout the process.

The final step to solve for 'x' in -14 = 12x is to get 'x' completely by itself. Right now, 'x' is being multiplied by 12. The opposite of multiplying by 12 is dividing by 12. So, we divide both sides of the equation by 12. This is our last balancing act! We get -14 / 12 = (12x) / 12. On the left side, -14 divided by 12 simplifies. Both 14 and 12 are divisible by 2. So, -14 / 2 = -7 and 12 / 2 = 6. This gives us -7/6. On the right side, 12x divided by 12 simply leaves us with 'x'. So, our solution is x = -7/6. Woohoo! We found it! It's important to express the answer in its simplest fractional form, which we've done here. Sometimes, you might be asked to provide a decimal answer, in which case -7/6 is approximately -1.167. Always check the instructions on how to present your final answer. This final division step utilizes the multiplicative inverse property, completing the isolation of the variable.

Verifying Your Solution

Now, the really cool part, guys, is checking our work. It's always a good idea to plug our answer back into the original equation to make sure it's correct. Our original equation was 3x + 9 = 15x + 23, and we found that x = -7/6. Let's substitute -7/6 for 'x' on both sides and see if they equal each other. This verification process is crucial because it builds confidence in your answer and helps catch any silly mistakes you might have made along the way. It's the ultimate test of whether your algebraic maneuvering was spot on. Think of it as a final quality check before declaring victory.

On the left side: 3x + 9 becomes 3 * (-7/6) + 9. First, multiply 3 by -7/6. That's (3 * -7) / 6 = -21/6. This fraction can be simplified by dividing both numerator and denominator by 3, giving us -7/2. Now, add 9. To add -7/2 and 9, we need a common denominator. We can write 9 as 18/2. So, -7/2 + 18/2 = ( -7 + 18 ) / 2 = 11/2. That's our left side value.

Now, let's check the right side: 15x + 23 becomes 15 * (-7/6) + 23. Multiply 15 by -7/6. That's (15 * -7) / 6 = -105/6. Both 105 and 6 are divisible by 3. So, -105 / 3 = -35 and 6 / 3 = 2. This gives us -35/2. Now, add 23. To add -35/2 and 23, we need a common denominator. Write 23 as 46/2. So, -35/2 + 46/2 = ( -35 + 46 ) / 2 = 11/2.

Since the left side (11/2) equals the right side (11/2), our solution x = -7/6 is correct! It's such a good feeling when your check works out, right? This verification step is arguably as important as solving the equation itself, as it confirms the accuracy of your efforts and reinforces the understanding of how variables and constants interact within an equation. It’s the final stamp of approval on your mathematical journey.

Common Mistakes to Avoid

Alright, let's talk about some common pitfalls when you're solving equations like 3x + 9 = 15x + 23. One of the biggest mistakes guys make is with sign errors. When you move a number or a variable term to the other side of the equation, you must change its sign. Forgetting to do this, or accidentally flipping the sign when you didn't mean to, can throw off your entire answer. For instance, if you subtract 3x from both sides, make sure you're correctly calculating 15x - 3x and not, say, 15x + 3x. Always double-check the signs after each step, especially when dealing with negative numbers.

Another frequent issue is arithmetic mistakes. Simple addition, subtraction, multiplication, or division errors can happen to anyone, especially when working with fractions or negative numbers. Remember that checking your work, as we did with plugging the solution back in, is the best defense against these kinds of errors. Take your time with the calculations. If you're unsure about a specific calculation, like adding fractions with different denominators, it might be worth doing that part separately or using a calculator if allowed, just to be sure. Accuracy in arithmetic is the bedrock of correct algebraic solutions.

Finally, some people get confused about when to add/subtract and when to multiply/divide. Remember the order: First, you generally want to group your variable terms and constant terms by adding or subtracting. Only after you've isolated the variable term (like having '12x' on one side) do you then use multiplication or division to get the variable completely by itself. Trying to divide too early can make the equation much messier. Keep the goal in mind: isolate the variable step-by-step. Understanding this sequence is key to navigating algebraic equations efficiently and accurately.

Why Solving Equations Matters

So, why do we even bother learning to solve equations like 3x + 9 = 15x + 23? It might seem like just another math problem, but honestly, this skill is super useful in tons of real-world situations, even if you don't realize it. Think about budgeting – you might have a fixed income and some variable expenses, and you need to figure out how much you can spend on other things. That's essentially an equation! Or maybe you're planning a trip and want to know how many days you can afford to stay based on your daily budget and total savings. Algebra helps you model these scenarios and find the unknown quantity, which is 'x' in our case.

Beyond personal finance, algebraic thinking is crucial in science, engineering, economics, and even computer programming. When scientists are trying to understand a phenomenon, they often create mathematical models using equations to represent relationships between different variables. Engineers use equations to design everything from bridges to smartphones. Economists use them to predict market trends. Programmers use them to create algorithms that power the apps and websites we use every day. The ability to break down a problem, represent it with an equation, and solve for the unknown is a fundamental skill that opens doors to many exciting fields. It's all about logical reasoning and problem-solving, which are valuable no matter what career path you choose.

Learning to solve these basic equations builds a strong foundation for tackling more complex mathematical concepts later on. It trains your brain to think logically, systematically, and critically. Each equation you solve is like a small workout for your mind, strengthening your analytical abilities. So, next time you're working on an algebra problem, remember that you're not just finding a number; you're honing skills that will serve you well throughout your life. It's about empowering yourself with the tools to understand and navigate the world around you more effectively. Keep practicing, and you'll be amazed at what you can solve!