Solving For X: A Step-by-Step Guide
Hey guys! Let's dive into solving a classic algebraic equation. Today, we're tackling the problem: 2 = (3x + 17) / 4. Don't worry; it's not as scary as it looks! We'll break it down step-by-step, so you can confidently solve for x. Understanding how to isolate variables is a fundamental skill in algebra, and this example will give you a solid foundation. We’ll go through the process slowly and carefully, making sure each step is clear and easy to follow. Mastering these kinds of equations opens the door to more complex mathematical problems and real-world applications. Stick with me, and by the end of this guide, you'll be solving equations like a pro! Remember, practice makes perfect, so grab a pen and paper, and let's get started on this mathematical adventure together. This isn’t just about getting the right answer; it’s about understanding the process of solving for x. It's about building a logical approach to problem-solving that you can apply to all sorts of challenges, both inside and outside the classroom.
1. Understanding the Equation
Before we jump into the solution, let's make sure we understand what the equation 2 = (3x + 17) / 4 is telling us. This equation states that the expression (3x + 17) divided by 4 is equal to 2. Our goal is to find the value of x that makes this statement true. Think of it as a puzzle where x is the missing piece. To find it, we need to isolate x on one side of the equation. This means we want to manipulate the equation using mathematical operations until we have x all by itself on one side, and a numerical value on the other side. This value will be our solution. The equation itself is a balanced statement, meaning that both sides are equal. Any operation we perform must maintain this balance. If we add something to one side, we must add the same thing to the other. If we multiply one side by a number, we must multiply the other side by the same number. This principle of balance is the key to solving algebraic equations. We'll be using it repeatedly throughout the process, so make sure you grasp this fundamental concept. Now, let's move on to the first step in our solution: getting rid of the fraction.
2. Eliminating the Fraction
The first thing we want to do is get rid of that fraction. Fractions can make equations look more complicated than they are, so eliminating them is a good first step. In our equation, 2 = (3x + 17) / 4, the entire expression (3x + 17) is being divided by 4. To undo this division, we need to perform the opposite operation: multiplication. We're going to multiply both sides of the equation by 4. This is crucial – remember the balance we talked about earlier? What we do to one side, we must do to the other. So, we have:
4 * 2 = 4 * [(3x + 17) / 4]
On the left side, 4 multiplied by 2 is simply 8. On the right side, the multiplication by 4 cancels out the division by 4. This is the magic of inverse operations! They undo each other. So, we're left with:
8 = 3x + 17
See how much simpler the equation looks now? We've successfully eliminated the fraction and are one step closer to isolating x. This step is a common strategy in solving equations, especially those involving fractions. It clears the way for easier manipulation and helps us focus on the core variables. Now that we have a cleaner equation, let's move on to the next step: isolating the term with x.
3. Isolating the Term with x
Our equation is now 8 = 3x + 17. Our next goal is to isolate the term with x, which in this case is 3x. This means we need to get rid of the + 17 that's on the same side of the equation. To do this, we'll use another inverse operation: subtraction. Since 17 is being added to 3x, we'll subtract 17 from both sides of the equation. Again, maintaining balance is key! So, we have:
8 - 17 = 3x + 17 - 17
On the left side, 8 minus 17 is -9. On the right side, +17 and -17 cancel each other out, leaving us with just 3x. Our equation now looks like this:
-9 = 3x
We're getting closer! We've successfully isolated the term with x on one side of the equation. Now, we just need to get x completely by itself. To do that, we need to deal with the 3 that's being multiplied by x. This brings us to the final step: solving for x.
4. Solving for x
We're at the home stretch! Our equation is -9 = 3x. To get x by itself, we need to undo the multiplication by 3. The inverse operation of multiplication is division, so we'll divide both sides of the equation by 3. You guessed it – we're still keeping that balance in mind! So, we have:
-9 / 3 = (3x) / 3
On the left side, -9 divided by 3 is -3. On the right side, the division by 3 cancels out the multiplication by 3, leaving us with just x. Our equation now looks like this:
-3 = x
Or, to put it the other way around:
x = -3
We did it! We've successfully solved for x. The value of x that makes the original equation true is -3. This is our solution. It's important to remember that solving for a variable is a process of unraveling the equation, step by step, using inverse operations and always maintaining balance. And it is really important that each step is well understood. Let’s not move on if something is still confusing to you. It’s much better to ask for clarification or to review the previous sections before proceeding. The ultimate goal isn’t just about finding the answer to a single problem. It’s about developing a deep and flexible understanding of mathematics. This understanding will allow you to approach a wide range of problems with confidence.
5. Checking Our Solution
It's always a good idea to check your solution to make sure you haven't made any mistakes. To do this, we'll substitute our value for x (-3) back into the original equation: 2 = (3x + 17) / 4. So, we replace x with -3:
2 = [3(-3) + 17] / 4
Now, we simplify the right side of the equation. First, we multiply 3 by -3:
2 = [-9 + 17] / 4
Next, we add -9 and 17:
2 = 8 / 4
Finally, we divide 8 by 4:
2 = 2
The left side of the equation equals the right side of the equation! This confirms that our solution, x = -3, is correct. Checking your solution is a crucial step in problem-solving. It gives you confidence in your answer and helps you catch any errors you might have made along the way. It's like having a built-in safety net. Always take the time to check your work, especially in mathematics. It's a habit that will serve you well throughout your mathematical journey. If the check doesn't work out, it means you've made a mistake somewhere in your calculations, and you need to go back and review your steps. This process of checking and correcting is a valuable learning experience in itself.
Conclusion
Awesome! You've successfully solved for x in the equation 2 = (3x + 17) / 4. We took it step-by-step, eliminating the fraction, isolating the term with x, and finally, solving for x. We also learned the importance of checking our solution to ensure accuracy. Remember, solving equations is a fundamental skill in algebra. The techniques we used today can be applied to a wide variety of problems. The key is to understand the underlying principles: using inverse operations to undo mathematical operations and maintaining balance on both sides of the equation. Practice is key to mastering these skills. The more equations you solve, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and keep challenging yourself. Mathematics is a fascinating subject, and with a little effort, you can achieve great things. Don’t be afraid to make mistakes; they are valuable learning opportunities. When you encounter a challenging problem, break it down into smaller, more manageable steps. And remember, there are many resources available to help you, including textbooks, online tutorials, and your teachers and classmates. Keep up the great work, and I'll see you in the next mathematical adventure! And just a little reminder, these principles of algebra aren’t confined to the classroom. They’re incredibly useful in the real world too. Whether you’re calculating discounts while shopping, planning a budget, or even figuring out how long it will take to drive somewhere, algebra is your friend.