Let's break down how to solve this math problem step by step. We're given two equations: 2b = 273 and ab = 24, and our mission is to find the value of a + b. Grab your thinking caps, guys, because we're about to dive in!

Understanding the Problem

Before we start crunching numbers, it's super important to understand what the problem is asking. We have two unknowns, a and b, and two equations relating them. This means we can solve for the individual values of a and b, and then simply add them together to get our final answer. Math can be fun. The key is to stay organized and take it one step at a time.

Given Equations:

1. 2b = 273
2. ab = 24

Goal:

Find the value of a + b.

Step-by-Step Solution

Step 1: Solve for 'b' from the First Equation

The first equation is 2b = 273. To isolate b, we need to divide both sides of the equation by 2. This will give us the value of b.

2b / 2 = 273 / 2

b = 136.5

So, we've found that b equals 136.5. That was pretty straightforward, right?

Step 2: Substitute 'b' into the Second Equation

Now that we know the value of b, we can substitute it into the second equation, which is ab = 24. Replacing b with 136.5, we get:

a * 136.5 = 24

Step 3: Solve for 'a'

To find the value of a, we need to isolate it. We can do this by dividing both sides of the equation by 136.5:

a = 24 / 136.5

a ≈ 0.1758

So, a is approximately 0.1758. We now have the values for both a and b.

Step 4: Calculate 'a + b'

The final step is to add the values of a and b together:

a + b = 0.1758 + 136.5

a + b ≈ 136.6758

Therefore, a + b is approximately 136.6758.

Verification

To make sure our answer is correct, let's plug the values of a and b back into the original equations:

1. 2b = 273

   2 * 136.5 = 273 (Correct)

2. ab = 24

   0.1758 * 136.5 ≈ 24 (Approximately correct, due to rounding)

Since both equations hold true (or are very close, considering rounding), we can be confident that our solution is correct.

Alternative Approaches

While the step-by-step method is clear and easy to follow, there might be other ways to approach this problem, especially if you're looking for shortcuts or have a strong algebraic intuition. However, for clarity and accuracy, the above method is generally the most reliable.

Practical Applications

Okay, so you might be thinking, "When am I ever going to use this in real life?" Well, these types of algebraic problems are the building blocks for many real-world applications. For example:

• Engineering: Engineers use systems of equations to design structures, circuits, and machines.
• Economics: Economists use equations to model markets and predict economic trends.
• Computer Science: Programmers use algebraic principles to develop algorithms and solve computational problems.
• Finance: Financial analysts use equations to calculate investments, manage risk, and forecast profits.

So, while the specific problem might seem abstract, the underlying skills are incredibly valuable in a wide range of fields.

Common Mistakes to Avoid

When solving problems like this, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

• Arithmetic Errors: Double-check your calculations, especially when dividing or multiplying decimals.
• Incorrect Substitution: Make sure you're substituting the correct values into the correct equations.
• Rounding Errors: Be mindful of rounding, especially if you're working with decimals. Round too early, and your final answer might be significantly off.
• Algebraic Mistakes: Ensure you're following the correct order of operations and applying algebraic rules correctly.

Tips for Success

Here are a few tips to help you succeed in solving algebraic problems:

• Stay Organized: Keep your work neat and organized. This will help you avoid mistakes and make it easier to track your progress.
• Show Your Work: Don't skip steps. Showing your work makes it easier to identify and correct errors.
• Check Your Answers: Always check your answers by plugging them back into the original equations.
• Practice Regularly: The more you practice, the better you'll become at solving algebraic problems.
• Understand the Concepts: Don't just memorize formulas. Make sure you understand the underlying concepts.

Conclusion

So, to recap, if 2b = 273 and ab = 24, then a + b is approximately 136.6758. We solved this problem by first finding the value of b, then substituting it into the second equation to find the value of a, and finally adding a and b together. Remember to double-check your work and avoid common mistakes, and you'll be well on your way to mastering algebra!

Key Takeaways:

• Isolate Variables: Solve for one variable at a time.
• Substitute Carefully: Substitute values accurately to avoid errors.
• Check Your Work: Always verify your solution.

Keep practicing, and you'll become a pro at solving these types of problems in no time! Remember, math isn't about getting the right answer; it's about the process of learning and problem-solving. Embrace the challenge, and have fun with it!

Additional Practice Problems

Want to test your skills further? Here are a few more practice problems similar to the one we just solved:

1. If 3x = 45 and xy = 15, find x + y.
2. If 5p = 125 and pq = 25, find p + q.
3. If 4m = 64 and mn = 32, find m + n.

Try solving these on your own, and feel free to share your solutions in the comments below! And if you get stuck, don't worry – just review the steps we outlined earlier, and you'll be able to figure it out. Keep going guys!

The Importance of Algebra

Algebra isn't just about solving equations; it's a way of thinking. It teaches you how to break down complex problems into smaller, more manageable parts, how to identify patterns, and how to think logically. These skills are valuable not only in math and science but also in everyday life.

Algebra helps us develop critical thinking, problem-solving, and analytical skills. From balancing your budget to planning a road trip, algebraic thinking is always at play. It's a fundamental skill that empowers us to make informed decisions and navigate the world around us more effectively.

Final Thoughts

So, whether you're a student struggling with algebra or someone who just wants to brush up on their math skills, remember that practice makes perfect. Don't be afraid to ask for help when you need it, and always strive to understand the underlying concepts. With a little effort and perseverance, you can conquer any math problem that comes your way. Stay curious, keep learning, and never stop exploring the fascinating world of mathematics!

And there you have it! We've successfully solved for a + b given the equations 2b = 273 and ab = 24. I hope this comprehensive guide has been helpful. Keep practicing, and you'll become a math whiz in no time. Good luck, and happy problem-solving!