Solve The Algebra Problem The Triple Of A Number Increased By 58

Hey there, math enthusiasts! Let's dive into a fascinating mathematical puzzle today. We're going to unravel the mystery behind the statement: "The triple of a number increased by 58 is 106. What is the number?" This type of problem is a classic example of an algebraic equation, and by breaking it down step-by-step, we'll see how straightforward it can be to solve. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, it's crucial to understand what the problem is actually asking. Let's dissect the statement piece by piece:

• "The triple of a number": This means we have a number, which we don't know yet, and we're multiplying it by 3. In algebra, we often represent unknown numbers with variables, like x. So, "the triple of a number" can be written as 3x.
• "increased by 58": This indicates that we're adding 58 to the previous expression. So now we have 3x + 58.
• "is 106": This tells us that the entire expression we've built up is equal to 106. Therefore, we can write the complete equation as 3x + 58 = 106.

Now that we've translated the words into a mathematical equation, we have a clear roadmap for finding the unknown number. The beauty of algebra lies in its ability to turn complex word problems into manageable equations. By understanding the language of mathematics, we can unlock the solutions hidden within these puzzles.

Setting Up the Equation

Alright, guys, now that we've dissected the problem, let's formally set up the equation. As we discussed, the statement "The triple of a number increased by 58 is 106" translates directly into the algebraic equation:

3x + 58 = 106

Here, x represents the unknown number we're trying to find. The left side of the equation, 3x + 58, represents the "triple of a number increased by 58," and the right side, 106, represents the result. Our goal now is to isolate x on one side of the equation, which will reveal its value. Think of it like a balancing scale: we need to perform the same operations on both sides to maintain the balance and keep the equation true.

Setting up the equation correctly is the most crucial step in solving any algebraic problem. It's like laying the foundation for a building; if the foundation is shaky, the entire structure might crumble. So, always take your time to carefully translate the words into mathematical symbols and ensure that the equation accurately represents the problem.

Solving the Equation Step-by-Step

Okay, let's get down to the nitty-gritty and solve this equation! Remember, our aim is to isolate x on one side. We'll do this by performing inverse operations – operations that undo each other – on both sides of the equation. Here's how it goes:

1. Subtract 58 from both sides: This will get rid of the +58 on the left side. Remember, what we do to one side, we must do to the other to keep the equation balanced.

   3x + 58 - 58 = 106 - 58

   This simplifies to:

   3x = 48

2. Divide both sides by 3: Now, we need to get rid of the 3 that's multiplying x. To do this, we divide both sides by 3.

   (3x) / 3 = 48 / 3

   This simplifies to:

   x = 16

And there we have it! We've successfully isolated x and found its value. The number we were looking for is 16. Isn't it satisfying when the puzzle pieces fall into place? This step-by-step approach is the key to solving most algebraic equations. By carefully applying inverse operations, we can systematically unravel the unknown.

Verifying the Solution

Awesome! We've found that x = 16, but before we declare victory, it's always a good idea to verify our solution. This ensures that we haven't made any mistakes along the way. To verify, we simply substitute the value we found for x back into the original equation:

3x + 58 = 106

Substitute x = 16:

3 * (16) + 58 = 106

Now, let's simplify:

48 + 58 = 106

106 = 106

Ta-da! The equation holds true. This confirms that our solution, x = 16, is correct. Verifying your solution is like double-checking your work before submitting it. It gives you confidence that you've got the right answer and helps catch any potential errors. So, never skip this crucial step!

Alternative Approaches to Solving

While we've solved the equation using the standard algebraic method, it's always cool to explore alternative approaches. Sometimes, a different perspective can make the solution even clearer, or it can be helpful if you get stuck using the usual method. Here are a couple of alternative ways we could have tackled this problem:

1. Working Backwards: Instead of setting up an equation, we could have worked backwards from the final result. We know that the triple of a number increased by 58 is 106. So, we could first subtract 58 from 106, which gives us 48. This tells us that the triple of the number is 48. Then, we could divide 48 by 3 to find the number itself, which is 16.

2. Trial and Error (with a Twist): While trial and error might seem like a less sophisticated method, it can be effective, especially if you approach it strategically. We could start by guessing a number and plugging it into the original statement. If the result is too high or too low, we can adjust our guess accordingly. For example, if we guessed 20, the triple would be 60, and adding 58 would give us 118, which is too high. This tells us that the number must be less than 20. We could then try a smaller number, like 15, and so on, until we find the correct answer. The "twist" here is to use the results of our guesses to inform our next guess, making the process more efficient.

Exploring alternative approaches not only gives you more tools in your problem-solving arsenal but also deepens your understanding of the underlying concepts. It's like learning different routes to the same destination; the more routes you know, the better equipped you are to navigate any situation.

Real-World Applications of Algebra

Guys, you might be wondering, "Okay, this is cool, but when am I ever going to use this in the real world?" Well, the truth is, algebra is everywhere! It's not just about solving equations with x and y; it's a powerful tool for modeling and solving problems in various aspects of life. Let's explore some real-world applications of algebra:

• Finance: Algebra is essential for managing money, budgeting, and making financial decisions. For example, you can use algebraic equations to calculate interest on loans, determine the monthly payments on a mortgage, or plan for retirement savings.
• Science and Engineering: Algebra is the foundation of many scientific and engineering disciplines. Scientists use algebraic equations to model physical phenomena, analyze data, and make predictions. Engineers use algebra to design structures, circuits, and machines.
• Computer Programming: Algebra is crucial for writing computer code. Programmers use variables, equations, and functions to create algorithms and solve computational problems.
• Everyday Life: Even in everyday situations, we often use algebraic thinking without realizing it. For example, when we're cooking and need to adjust a recipe for a different number of servings, we're using proportions, which are algebraic relationships. When we're planning a road trip and calculating how long it will take to reach our destination, we're using the formula distance = speed * time, which is an algebraic equation.

Algebra provides a framework for logical thinking and problem-solving that extends far beyond the classroom. By mastering algebraic concepts, you're equipping yourself with valuable skills that will benefit you in countless ways throughout your life.

Common Mistakes to Avoid

Alright, let's talk about common mistakes. We all make them, especially when we're learning something new. The key is to recognize them and learn how to avoid them. Here are a few pitfalls to watch out for when solving algebraic equations like the one we tackled today:

1. Incorrectly Applying Inverse Operations: This is a big one. Remember, the goal is to isolate the variable by performing the opposite operation. If you're adding, you need to subtract; if you're multiplying, you need to divide. Make sure you're applying the correct inverse operation to both sides of the equation.

2. Forgetting to Perform Operations on Both Sides: This is crucial for maintaining the balance of the equation. Whatever you do to one side, you must do to the other. If you only subtract 58 from the left side, for example, the equation will no longer be true.

3. Making Arithmetic Errors: Simple calculation mistakes can throw off your entire solution. Double-check your arithmetic, especially when dealing with negative numbers or fractions.

4. Misinterpreting the Problem: Before you even start solving, make sure you fully understand what the problem is asking. Read the problem carefully and identify the key information.

5. Skipping the Verification Step: As we discussed earlier, verifying your solution is essential for catching errors. Don't skip this step!

By being aware of these common mistakes, you can actively work to avoid them and improve your problem-solving skills. Remember, mistakes are opportunities for learning and growth.

Practice Problems for You

Okay, guys, now it's your turn to shine! The best way to master algebra is through practice. So, I've whipped up a few similar problems for you to try. Grab a pencil and paper, and let's put those skills to the test!

1. The double of a number decreased by 15 is 45. What is the number?
2. Five times a number increased by 8 is 53. What is the number?
3. The sum of a number and its half is 24. What is the number?

Try solving these problems using the same step-by-step approach we used earlier. Remember to set up the equation, solve for the unknown variable, and verify your solution. And don't be afraid to try different approaches if you get stuck!

If you're feeling extra confident, try creating your own word problems and solving them. This is a great way to deepen your understanding of algebraic concepts.

Conclusion

Wrapping things up, guys! We've successfully decoded the mystery of "the triple of a number increased by 58 is 106." We've seen how to translate word problems into algebraic equations, solve those equations step-by-step, and verify our solutions. We've also explored alternative approaches and discussed the real-world applications of algebra. Remember, algebra is more than just a bunch of symbols and equations; it's a powerful tool for problem-solving and logical thinking.

So, keep practicing, keep exploring, and keep those mathematical gears turning! You've got this!

What number, when tripled and increased by 58, equals 106?

Solve the Algebra Problem The Triple of a Number Increased by 58