Solving The Equation 3x/4 = 2x + 5 A Step-by-Step Guide
Hey guys! Today, we're diving deep into a math problem from the IFSC 2018 exam. We're going to break down the equation 3x/4 = 2x + 5 step-by-step, making sure everyone understands how to solve it. Math can seem intimidating, but with a clear approach, even the trickiest equations become manageable. So, let's put on our thinking caps and get started!
Decoding the Equation IFSC 2018
At the heart of our mathematical journey today lies the equation 3x/4 = 2x + 5. This seemingly simple algebraic expression holds a wealth of mathematical concepts within it, waiting to be unlocked. Before we jump into solving it, let's take a moment to appreciate the beauty and power of algebra. Algebra, at its core, is a language – a language of symbols and rules that allows us to express relationships between quantities and solve for unknowns. In this case, our unknown is represented by the variable 'x', and our mission is to find the value (or values) of 'x' that make the equation true. But why is this important? Why do we spend time wrestling with equations like this? Well, the ability to solve algebraic equations is a fundamental skill that permeates nearly every aspect of our lives, from calculating the tip at a restaurant to designing complex engineering systems. Equations like 3x/4 = 2x + 5 may seem abstract, but they are the building blocks of mathematical models that help us understand and interact with the world around us. So, as we embark on this journey to solve this equation, let's keep in mind the broader context of why we're doing it – we're not just manipulating symbols; we're developing a powerful tool for problem-solving and critical thinking.
Now, let's dig into the specifics of our equation. Notice the different components: We have a fraction on one side (3x/4), a term with 'x' on the other side (2x), and a constant term (+5). Each of these components plays a crucial role in the equation, and our task is to manipulate them in a way that isolates 'x' and reveals its value. To do this effectively, we'll need to employ a combination of algebraic techniques, including clearing fractions, combining like terms, and using inverse operations. Don't worry if these terms sound intimidating right now; we'll break them down step-by-step as we go through the solution process. The key is to approach the equation systematically, with a clear understanding of the rules and principles of algebra. So, let's take a deep breath, sharpen our pencils, and get ready to decode the equation 3x/4 = 2x + 5!
Step-by-Step Solution of IFSC 2018 Equation
Okay, guys, let's roll up our sleeves and get into the nitty-gritty of solving 3x/4 = 2x + 5. The first thing we want to tackle is that fraction, because fractions can sometimes make things look a bit more complicated than they really are. Our goal is to clear the fraction, and the way we do that is by multiplying both sides of the equation by the denominator, which in this case is 4. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced. It's like a mathematical seesaw – if you add weight to one side, you have to add the same weight to the other side to keep it level.
So, we multiply both sides of the equation by 4:
4 * (3x/4) = 4 * (2x + 5)
On the left side, the 4 in the numerator cancels out the 4 in the denominator, leaving us with just 3x. On the right side, we need to distribute the 4 to both terms inside the parentheses. This means we multiply 4 by 2x and 4 by 5. This distributive property is a crucial tool in algebra, and it's important to get comfortable using it. It allows us to simplify expressions that involve parentheses by multiplying the term outside the parentheses by each term inside. So, let's apply the distributive property and see what we get:
3x = 8x + 20
Now our equation looks much cleaner and simpler! We've successfully cleared the fraction, and we're one step closer to isolating 'x'. The next step is to gather all the terms with 'x' on one side of the equation. To do this, we'll subtract 8x from both sides. Again, remember the seesaw analogy – we're doing the same thing to both sides to maintain balance. Subtracting 8x from both sides gives us:
3x - 8x = 8x + 20 - 8x
On the left side, 3x minus 8x is -5x. On the right side, the 8x and -8x cancel each other out, leaving us with just 20. So now our equation looks like this:
-5x = 20
We're almost there! We've got all the 'x' terms on one side and the constant term on the other side. The last step is to isolate 'x' completely. Right now, 'x' is being multiplied by -5. To undo this multiplication, we need to divide both sides of the equation by -5. This is another example of using an inverse operation – we're using division to undo multiplication. So, let's divide both sides by -5:
-5x / -5 = 20 / -5
On the left side, the -5s cancel out, leaving us with just 'x'. On the right side, 20 divided by -5 is -4. So, finally, we have our solution:
x = -4
Woohoo! We did it! We've successfully solved the equation 3x/4 = 2x + 5 and found that x = -4. But our job isn't quite done yet. It's always a good idea to check our answer to make sure we haven't made any mistakes along the way.
Verifying the Solution of IFSC 2018
Alright, mathletes, we've arrived at a solution, x = -4, but like any good detectives, we need to verify our findings! Plugging our solution back into the original equation is like checking our alibi – it ensures that our answer holds up under scrutiny. This step is crucial because it helps us catch any errors we might have made during the solving process. Sometimes, a simple mistake in arithmetic can lead to a wrong answer, and verifying our solution is the best way to catch these mistakes before they become a problem.
So, let's take our solution, x = -4, and substitute it back into the original equation:
3x/4 = 2x + 5
Replacing 'x' with -4, we get:
3*(-4)/4 = 2*(-4) + 5
Now, let's simplify both sides of the equation separately. On the left side, we have 3 times -4, which is -12. Then we divide -12 by 4, which gives us -3. So the left side simplifies to -3.
On the right side, we have 2 times -4, which is -8. Then we add 5 to -8, which gives us -3. So the right side also simplifies to -3.
Now we can see if both sides of the equation are equal:
-3 = -3
Eureka! The left side equals the right side! This confirms that our solution, x = -4, is indeed correct. We've successfully verified our answer, and we can be confident that we've solved the equation accurately. This process of verification is a valuable skill in mathematics and in life in general. It teaches us the importance of checking our work, being thorough, and ensuring that our solutions are valid. So, always remember to verify your solutions whenever possible – it's a great way to build confidence in your mathematical abilities and avoid making careless errors.
By verifying our solution, we've not only confirmed that we have the correct answer, but we've also reinforced our understanding of the equation and the steps we took to solve it. This deeper understanding will be invaluable as we tackle more complex mathematical problems in the future. So, congratulations on solving and verifying the equation 3x/4 = 2x + 5! You've demonstrated your problem-solving skills and your commitment to accuracy. Keep up the great work!
Key Takeaways for IFSC 2018 and Beyond
Okay, team, we've conquered the equation 3x/4 = 2x + 5! But solving a single problem is just the beginning. The real power comes from understanding the concepts and strategies we used so we can apply them to a whole range of problems. Let's zoom out and look at the big picture, highlighting the key takeaways from this exercise. These takeaways aren't just about this specific equation; they're about building a solid foundation in algebra that will serve you well in the IFSC 2018 exam and beyond.
First and foremost, let's talk about clearing fractions. Fractions can often make equations look more complicated than they are, and clearing them is a powerful technique for simplification. We did this by multiplying both sides of the equation by the denominator of the fraction. This eliminates the fraction and transforms the equation into a more manageable form. Remember, the key is to multiply every term on both sides of the equation by the denominator. This ensures that we maintain the balance of the equation and don't change its fundamental meaning. Clearing fractions is a technique that you'll use again and again in algebra, so make sure you're comfortable with it. It's like having a secret weapon in your mathematical arsenal!
Next up is the distributive property. This property allows us to multiply a term by an expression inside parentheses. We used it when we multiplied 4 by (2x + 5). Remember, the distributive property states that a*(b + c) = ab + ac. This means we need to multiply the term outside the parentheses by each term inside the parentheses. It's like making sure everyone gets a fair share! The distributive property is another fundamental tool in algebra, and mastering it will help you simplify a wide variety of expressions and equations.
Another crucial takeaway is the importance of isolating the variable. Our ultimate goal in solving an equation is to get the variable (in this case, 'x') all by itself on one side of the equation. We do this by performing inverse operations. For example, to undo addition, we subtract; to undo multiplication, we divide. The key is to perform the same operation on both sides of the equation to maintain balance. Think of it like a tug-of-war – if you pull on one side, you need to pull equally on the other side to keep the rope from moving. Isolating the variable is like winning the tug-of-war – you've successfully separated 'x' from all the other terms and revealed its value.
And last but definitely not least, we have the verification step. Always, always, always check your solution! Substituting your solution back into the original equation is the best way to ensure that you haven't made any mistakes along the way. It's like proofreading your work before you submit it – it's a chance to catch any errors and make sure your answer is accurate. Verification not only gives you confidence in your solution, but it also reinforces your understanding of the equation and the solving process. It's a win-win situation!
So, there you have it – the key takeaways from solving the equation 3x/4 = 2x + 5. By mastering these concepts and strategies, you'll be well-equipped to tackle a wide range of algebraic problems, not just on the IFSC 2018 exam, but in all your future mathematical endeavors. Keep practicing, keep learning, and remember that math is a journey, not a destination. Enjoy the ride!
Practice Problems for Mastering Equations
Alright, champions, we've dissected the equation 3x/4 = 2x + 5 and extracted some golden nuggets of algebraic wisdom. Now it's time to put those nuggets to work! Practice is the secret sauce to mastering any skill, and algebra is no exception. Think of these practice problems as your training ground – a place to hone your skills, build your confidence, and solidify your understanding. The more you practice, the more comfortable and fluent you'll become in the language of algebra. So, let's grab our metaphorical weights and hit the algebraic gym!
I've whipped up a few practice problems that will challenge you to apply the techniques we've discussed, like clearing fractions, using the distributive property, isolating the variable, and of course, verifying your solutions. These problems are designed to build on what you've learned and push you to think critically and creatively. Remember, the goal isn't just to get the right answer; it's to understand the process of solving equations. So, take your time, show your work, and don't be afraid to make mistakes. Mistakes are just opportunities to learn and grow!
Here are a few problems to get you started:
- Solve for x: (2x + 1)/3 = x - 2
- Solve for y: 5y - 7 = 2y + 8
- Solve for a: (4a - 3)/2 = 3a + 1
- Solve for b: -2(b + 4) = 5b - 6
These problems cover a range of scenarios, from equations with fractions to equations involving the distributive property. As you work through them, pay attention to the steps you're taking and why you're taking them. Ask yourself questions like, "What's the best way to clear this fraction?" or "How can I isolate the variable in this equation?" The more you think critically about your approach, the better you'll become at problem-solving.
And remember, the most important step is to verify your solutions. Once you've found a solution, plug it back into the original equation to make sure it works. This is your safety net, your chance to catch any errors and ensure that your answer is correct. Verification is a habit that will serve you well in all your mathematical endeavors.
So, dive into these practice problems with enthusiasm and determination. Don't be discouraged if you get stuck – that's part of the learning process. If you're struggling with a particular problem, revisit the steps we discussed earlier, review your notes, or even ask a friend or teacher for help. The key is to keep practicing and keep learning. With each problem you solve, you'll be building your skills, your confidence, and your mathematical prowess. You've got this!
Final Thoughts and Encouragement for the IFSC 2018
Guys, we've reached the end of our journey through the equation 3x/4 = 2x + 5! We've not only solved the equation itself, but we've also explored the underlying concepts, strategies, and skills that make algebra so powerful. We've talked about clearing fractions, using the distributive property, isolating the variable, and the all-important step of verifying our solutions. We've even tackled some practice problems to solidify our understanding. Now, as we wrap things up, I want to leave you with a few final thoughts and words of encouragement, especially for those of you preparing for the IFSC 2018 exam.
First and foremost, remember that math is a skill, and like any skill, it improves with practice. The more you work with equations, the more comfortable and confident you'll become. Don't be afraid to challenge yourself, to tackle problems that seem difficult at first. The feeling of accomplishment you'll get when you finally crack a tough problem is incredibly rewarding. So, keep practicing, keep pushing yourself, and keep growing your mathematical abilities.
Another important thing to remember is that math is a process. It's not just about memorizing formulas and procedures; it's about understanding the underlying logic and reasoning. When you approach a problem, take the time to think it through, to understand the relationships between the different parts. Don't just blindly apply a formula; instead, ask yourself why the formula works and how it applies to the specific problem you're trying to solve. This deeper understanding will not only help you solve problems more effectively, but it will also make math more enjoyable and engaging.
And finally, remember that math is a tool. It's a tool that can help you understand the world around you, solve problems in your daily life, and achieve your goals. From calculating the tip at a restaurant to designing a bridge, math is everywhere. The skills you're developing in algebra are not just for exams; they're for life. So, embrace the power of math, and use it to make a positive impact on the world.
For those of you preparing for the IFSC 2018 exam, I want to offer a special word of encouragement. You've got this! You've put in the work, you've studied the concepts, and you've practiced the skills. Now it's time to trust in your preparation and go out there and do your best. Remember to stay calm, read each question carefully, and show your work. And don't forget to verify your solutions! You've got the knowledge and the skills to succeed. Believe in yourself, and go make it happen!
So, as we conclude our exploration of the equation 3x/4 = 2x + 5, I want to thank you for joining me on this mathematical adventure. I hope you've found it helpful, informative, and maybe even a little bit fun. Keep practicing, keep learning, and never stop exploring the wonderful world of mathematics. You've got the potential to achieve great things, so go out there and make it happen! Good luck, and happy solving!
