Solving 30x + 2 = 40x - 10 A Step-by-Step Guide

by Rajiv Sharma 48 views

Hey guys! Ever get stuck on a math problem and feel like you're just spinning your wheels? Well, today we're going to break down a classic algebraic equation step-by-step, so you can see exactly how to solve it. We're tackling the equation 30x + 2 = 40x - 10. Don't worry, it's not as scary as it looks! We'll go through each step nice and slow, making sure you understand the 'why' behind the 'how'. So, grab your pencils and let's dive in!

Understanding the Equation

Before we jump into the solution, let's first understand what the equation 30x + 2 = 40x - 10 actually represents. In simple terms, it's a mathematical statement asserting that two expressions, 30x + 2 and 40x - 10, are equal. The variable 'x' is an unknown quantity we aim to find. Think of 'x' as a mystery number we need to uncover. Our mission is to find the value of 'x' that makes the equation true. In other words, we want to find the number that, when substituted for 'x' on both sides of the equation, results in the same numerical value. To do this, we'll use the principles of algebra to isolate 'x' on one side of the equation. This involves performing operations on both sides of the equation to maintain the balance while simplifying it. The ultimate goal is to get 'x' all by itself on one side, revealing its true value. Remember, equations are like scales – whatever you do to one side, you must do to the other to keep them balanced. This fundamental concept guides every step we take in solving algebraic equations. So, with this understanding, let's move on to the first step in solving our equation!

Step 1: Grouping Like Terms

The secret to cracking any algebraic equation lies in organizing things neatly. Our first move in solving 30x + 2 = 40x - 10 is to gather all the terms containing 'x' on one side of the equation and all the constant terms (the numbers) on the other side. This is like sorting your socks – putting similar things together makes life easier! To do this, we can subtract 30x from both sides of the equation. Remember our scale analogy? What we do on one side, we must do on the other to keep things balanced. Subtracting 30x from both sides gives us: 30x + 2 - 30x = 40x - 10 - 30x. Now, simplify both sides. On the left, 30x - 30x cancels out, leaving us with just 2. On the right, 40x - 30x simplifies to 10x. Our equation now looks cleaner: 2 = 10x - 10. See? We're making progress! Next, we need to move the constant term (-10) from the right side to the left side. We can do this by adding 10 to both sides of the equation: 2 + 10 = 10x - 10 + 10. Simplifying again, we get 12 = 10x. Now, all our 'x' terms are on the right, and all our constant terms are on the left. We're one step closer to isolating 'x' and finding its value. Grouping like terms is a fundamental technique in algebra, and mastering it is crucial for solving more complex equations. So, let's move on to the next step and finish this puzzle!

Step 2: Isolating the Variable

Alright, we're on the home stretch! We've grouped our like terms, and our equation currently looks like this: 12 = 10x. Now, the final boss: isolating 'x'. Remember, our goal is to get 'x' all by itself on one side of the equation. Right now, 'x' is being multiplied by 10. To undo this multiplication, we need to perform the opposite operation: division. We'll divide both sides of the equation by 10. Why both sides? Because, you guessed it, we need to keep the equation balanced! Dividing both sides by 10 gives us: 12 / 10 = (10x) / 10. On the right side, the 10 in the numerator and the 10 in the denominator cancel each other out, leaving us with just 'x'. On the left side, 12 / 10 simplifies to 1.2. So, our equation now reads: 1.2 = x. Boom! We've done it! We've successfully isolated 'x' and found its value. This step, isolating the variable, is the climax of solving most algebraic equations. It's where all our hard work pays off and the mystery number is revealed. You've now mastered a key algebraic skill. Give yourself a pat on the back! But before we celebrate too much, let's take one final step to make sure our solution is correct.

Step 3: Verifying the Solution

Okay, we've found that x = 1.2. But how do we know if we're right? Math isn't about blindly following steps; it's about understanding and verifying. This is where the crucial step of verification comes in. To verify our solution, we'll substitute x = 1.2 back into our original equation: 30x + 2 = 40x - 10. Let's plug it in! On the left side, we have 30 * 1.2 + 2. This equals 36 + 2, which is 38. On the right side, we have 40 * 1.2 - 10. This equals 48 - 10, which is also 38. Guess what? Both sides of the equation are equal! 38 = 38. This means our solution, x = 1.2, is correct! We've successfully solved and verified our answer. Verification is like the final seal of approval on your math work. It gives you confidence that you've not only followed the steps correctly but also understood the underlying principles. It's a practice that will save you from making careless mistakes and help you develop a deeper understanding of algebra. So, always remember to verify your solutions! Now that we've verified our solution, let's recap the steps we took and discuss the key takeaways from this exercise.

Recap of Steps

Let's quickly run through what we did to solve 30x + 2 = 40x - 10:

  1. Understanding the Equation: We started by understanding what the equation meant – that two expressions are equal, and we need to find the value of 'x' that makes it true.
  2. Grouping Like Terms: We gathered all the 'x' terms on one side and the constant terms on the other side by adding or subtracting terms from both sides of the equation.
  3. Isolating the Variable: We isolated 'x' by dividing both sides of the equation by the coefficient of 'x'.
  4. Verifying the Solution: We substituted our solution back into the original equation to make sure both sides were equal.

These four steps form a solid framework for solving linear equations. Remember, the key is to maintain balance by performing the same operations on both sides of the equation. Grouping like terms simplifies the equation, isolating the variable reveals its value, and verifying the solution confirms our answer. By mastering these steps, you'll be well-equipped to tackle a wide range of algebraic problems. Solving equations is like building a house – each step is a crucial part of the foundation. Now that we've built our house, let's reflect on the bigger picture and discuss some strategies for tackling similar equations in the future.

Strategies for Similar Equations

So, you've conquered 30x + 2 = 40x - 10. Awesome! But what about other equations? The good news is, the same principles apply. Let's talk strategies for tackling similar equations. First, always remember the golden rule of equations: whatever you do to one side, you must do to the other. This is the foundation of all algebraic manipulations. Second, simplify before you solve. If there are parentheses or fractions in the equation, get rid of them first. Distribute any multiplication over parentheses and clear fractions by multiplying both sides of the equation by the least common multiple of the denominators. Third, group like terms early in the process. This will make the equation easier to handle and prevent mistakes. Fourth, isolate the variable systematically. Use inverse operations (addition/subtraction, multiplication/division) to peel away the layers surrounding the variable until it stands alone. Finally, always verify your solution. This is your safety net. It catches errors and reinforces your understanding. Remember, practice makes perfect. The more equations you solve, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're part of the learning process. Analyze your mistakes, understand why they happened, and learn from them. Algebra is a powerful tool, and mastering it will open doors to more advanced mathematical concepts. So, keep practicing, keep exploring, and keep challenging yourself. You've got this!

Conclusion

We've successfully solved the equation 30x + 2 = 40x - 10 and learned a ton along the way! We broke down each step, from understanding the equation to verifying our solution. We also discussed key strategies for tackling similar equations. Remember, algebra is like a puzzle – each piece fits together to reveal the solution. By understanding the underlying principles and practicing consistently, you can become a master puzzle solver! The key takeaways are grouping like terms, isolating the variable, and always verifying your answer. These steps will serve you well in all your algebraic adventures. Math might seem daunting at times, but breaking it down into manageable steps makes it much less intimidating. So, don't be afraid to dive in, explore, and ask questions. The more you engage with math, the more you'll discover its beauty and power. You've taken a great step today in boosting your algebra skills. Keep up the fantastic work, and you'll be conquering even more complex equations in no time! Now, go forth and solve!