Finding A, B, And C: A Math Problem Solved!

Hey math enthusiasts! Today, we're diving into a fun little problem where we need to find the values of a, b, and c. It's like a puzzle, and trust me, it's not as hard as it might seem at first glance. We're given some clues, and with a bit of cleverness, we can crack the code and find the answers. So, let's roll up our sleeves and get started! We'll use the information we have, a bit of algebra, and a sprinkle of logic to solve this mathematical mystery. Ready to become math detectives? Let's go!

Understanding the Problem: The Given Information

Alright, guys, let's break down what we know. We have three crucial pieces of information that are going to be our best friends in this quest. These are the foundations upon which we'll build our solution. Think of them as the key ingredients in a recipe. Without them, we're lost!

Firstly, we are told that a + b + c = 979. This is our grand total; the sum of all three numbers. Secondly, we're given a + b = 568. This tells us that the combined value of a and b is 568. Lastly, we have b + c = 752. This provides the combined value of b and c. These three equations are our compass, guiding us towards the answers. We'll skillfully use these facts to isolate each variable and find the value of a, b, and c. It's like having a treasure map, and each equation is a step closer to the treasure – the solutions to our problem. We need to remember these pieces of information as they're the building blocks we need to uncover the values of a, b, and c.

So, to recap, our starting point is:

• a + b + c = 979 (Equation 1)
• a + b = 568 (Equation 2)
• b + c = 752 (Equation 3)

Now, let's see how we can use these equations to find the values of a, b, and c. We'll use a series of substitutions and a bit of mental juggling, which is pretty fun when you get the hang of it. Remember, the goal is to make it manageable, step by step.

Solving for the Variables: Step-by-Step Guide

Now, for the exciting part – actually solving the equations! This is where we put on our detective hats and start working through the clues. We'll tackle this step-by-step, so hang in there, it's easier than you might think. We have three equations, and our aim is to find out what a, b, and c are worth. Let's see how.

Step 1: Finding the Value of c

We know from Equation 1 that a + b + c = 979, and from Equation 2 that a + b = 568. The trick here is to substitute the value of a + b from Equation 2 into Equation 1. So, instead of writing a + b, we can replace it with 568. This gives us: 568 + c = 979. To find c, we subtract 568 from both sides of the equation. This isolates c and tells us c = 979 - 568 = 411. Boom! We've found the value of c!

Step 2: Finding the Value of a

Next up, let's find the value of a. We know that b + c = 752 (Equation 3), and we have just found out that c = 411. So, we can substitute c in Equation 3 with its value. This results in: b + 411 = 752. To find b, we subtract 411 from both sides, giving us b = 752 - 411 = 341. Now, with the value of b known, we can head back to Equation 2 (a + b = 568) and substitute the value of b, which is 341. Hence, a + 341 = 568. Finally, subtract 341 from both sides to reveal that a = 568 - 341 = 227. We've cracked the code and found the value of a!

Step 3: Finding the Value of b

We previously determined the value of b in Step 2, where we found that b = 341. However, if we wanted to verify or find b through another route, we could use our initial equations, a + b = 568 where a is 227. Substituting the value of a (227), then the equation becomes 227 + b = 568. Therefore, to find b, we subtract 227 from both sides to reveal that b = 568 - 227 = 341. Another way to confirm the value of b would be to use b + c = 752 where c is 411. Substituting the value of c (411), then the equation becomes b + 411 = 752. Therefore, to find b, we subtract 411 from both sides to reveal that b = 752 - 411 = 341. So, we found the value of b. We used multiple approaches and the numbers are consistent.

Recap of Solutions

• a = 227
• b = 341
• c = 411

Verifying the Solution: Checking Our Answers

Awesome, we've found our answers! But before we call it a day, let's make sure our solutions are correct. It's always a good idea to double-check our work. This is like proofreading an essay – you want to make sure everything adds up and that there are no mistakes. We can do this by plugging our values of a, b, and c back into our original equations and see if everything works out.

Checking with Equation 1: a + b + c = 979

Let's substitute our values: 227 + 341 + 411 = 979. Does this equation hold true? Yes, it does! 227 + 341 + 411 = 979. This confirms that our solution for a, b, and c is correct.

Checking with Equation 2: a + b = 568

Substitute the values for a and b: 227 + 341 = 568. Does this equation hold true? Absolutely! 227 + 341 = 568. This is another piece of evidence supporting the accuracy of our answers.

Checking with Equation 3: b + c = 752

Finally, let's check with Equation 3: b + c = 752. Plugging in the values, we get 341 + 411 = 752. And indeed, 341 + 411 = 752. Hooray! All our equations hold true with the values of a, b, and c we found. Our answers are verified, and we can be confident that we've solved the problem correctly. Congrats, guys! You did it!

Conclusion: We Solved It!

We did it, guys! We successfully found the values of a, b, and c! Using a combination of the given equations and a little bit of algebraic manipulation, we were able to solve this math problem. Remember, the key is to break down the problem into smaller, manageable steps. We carefully used the information we had, made strategic substitutions, and verified our answers. This method can be applied to many other math problems, so remember the steps and practice them.

This kind of problem helps us understand how different numbers relate to each other and shows us the power of algebra. It's not just about finding answers; it's about learning how to think logically and solve problems systematically. So next time you encounter a similar problem, you'll know exactly what to do. Keep practicing, and you'll get better and better at solving these kinds of mathematical puzzles. Keep up the excellent work, and never be afraid to tackle a math problem! You've got this, and you are all set to be math whizzes! Keep practicing, and you will become even better!