Solve For A, B, C: A+b+c=13975, A+b=10684, B-c=4379

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Solve for a, b, c: a+b+c=13975, a+b=10684, b-c=4379

Alright, guys, let's dive into this math problem where we need to figure out the values of a, b, and c. We've got a system of equations, and our mission is to crack it! This is a classic algebra puzzle, and I'm here to guide you through each step so you can solve it like a pro.

Breaking Down the Equations

So, first things first, let's lay out the equations we're working with:

  1. a + b + c = 13975
  2. a + b = 10684
  3. b - c = 4379

These equations give us a relationship between a, b, and c. Our goal is to manipulate these equations in such a way that we can isolate each variable and find its value. We'll use substitution and simplification to get there. Stay with me, it's going to be fun! To kick us off, let’s try to leverage the information we have in the first two equations to find the value of 'c'. This should get our ball rolling and make the problem much easier to solve step by step. Remember, in math, every step forward, no matter how small, brings us closer to the final solution. Keep your focus and believe in your problem-solving abilities, and you'll surely conquer this mathematical challenge!

Solving for 'c'

We have two equations that involve a, b, and c. The first equation is a + b + c = 13975, and the second one is a + b = 10684. Notice that a + b appears in both equations. This is perfect for substitution!

Replace a + b in the first equation with its value from the second equation:

10684 + c = 13975

Now, isolate c by subtracting 10684 from both sides:

c = 13975 - 10684

c = 3291

Woo-hoo! We found the value of c. Now that we know c, we can use this information to find the value of b. This is where the third equation comes into play. By substituting the value of c into the third equation, we'll be one step closer to solving the entire system. Keep your spirits high and your focus sharp as we continue to unravel this mathematical puzzle. Remember, the key to success in math is to break down complex problems into smaller, manageable steps. So, let's keep moving forward with confidence and determination!

Solving for 'b'

Now that we know c = 3291, we can use the third equation, b - c = 4379, to find b. Substitute the value of c into the equation:

b - 3291 = 4379

Isolate b by adding 3291 to both sides:

b = 4379 + 3291

b = 7670

Awesome, we've found the value of b! With both b and c in hand, it's time to find the value of a. We'll use the second equation, a + b = 10684. This is where all our hard work starts to pay off. Each value we've found brings us closer to completing the puzzle. Remember to double-check your work as you proceed to ensure accuracy. This step is crucial because it confirms that we're on the right track and prevents any errors from propagating through the rest of the solution. Let's keep our momentum going and nail this problem!

Solving for 'a'

We know that b = 7670, so we can use the second equation, a + b = 10684, to find a. Substitute the value of b into the equation:

a + 7670 = 10684

Isolate a by subtracting 7670 from both sides:

a = 10684 - 7670

a = 3014

Fantastic! We've found the value of a. Now we have all the values: a = 3014, b = 7670, and c = 3291.

Verification

To make sure our solution is correct, let's plug these values back into the original equations and see if they hold true.

  1. a + b + c = 13975

    3014 + 7670 + 3291 = 13975

    13975 = 13975 (True)

  2. a + b = 10684

    3014 + 7670 = 10684

    10684 = 10684 (True)

  3. b - c = 4379

    7670 - 3291 = 4379

    4379 = 4379 (True)

All three equations hold true with our values. That confirms that our solution is correct! This step is super important because it ensures that we haven't made any mistakes along the way. Nothing feels better than knowing you've nailed a problem, right? Plus, verification helps build confidence in your problem-solving skills. So, always take the time to double-check your work. Trust me, it's worth it!

Final Answer

So, the values are:

  • a = 3014
  • b = 7670
  • c = 3291

We did it! We successfully found the values of a, b, and c that satisfy all three equations. Great job, guys! This was a classic example of solving a system of equations using substitution and simplification. Remember, the key is to break down the problem into manageable steps and verify your solution to ensure accuracy. Keep practicing, and you'll become a master at solving these types of problems. You've got this!

Key Takeaways

To wrap things up, here are the key strategies we used to solve this system of equations:

  • Substitution: We replaced expressions in one equation with their equivalent values from another equation.
  • Simplification: We simplified equations by combining like terms and isolating variables.
  • Verification: We checked our solution by plugging the values back into the original equations.

By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems. And remember, math is not just about finding the right answers; it's about developing problem-solving skills that can be applied to various aspects of life. So, keep exploring, keep learning, and never stop challenging yourself. Happy solving!