Solving For A+b+c Given Two Equations

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Hey everyone! Today, we're diving into an interesting math problem. We are given two equations: 3a + 5b + 4c = 90 and 5a + 3b + 4c = 70. Our mission? To figure out the value of a + b + c. Sounds like a fun challenge, right? Let's break it down step by step. This article will explore the methods to tackle this problem and similar algebraic puzzles. We will focus on providing clear, understandable explanations to help anyone follow along, whether you're a math whiz or just brushing up on your algebra skills. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's really understand what we're dealing with. We have two equations, and these equations share three variables: a, b, and c. This is a classic setup for a system of linear equations. The trick here isn't to solve for each variable individually (though that's possible!), but to find the combined value of a + b + c. Recognizing this is crucial because it can lead us to a more efficient solution. Sometimes in math, the direct route isn't always the quickest! We're looking for a clever way to combine our equations to get exactly what we need.

The key to solving this problem lies in recognizing that we don't necessarily need to find the individual values of a, b, and c. Instead, we can manipulate the given equations to directly find the value of a + b + c. This involves using techniques like adding or subtracting the equations to eliminate terms and simplify the problem. It's like a mathematical puzzle where we rearrange the pieces to reveal the answer. Keep this in mind as we proceed – it's the core strategy for our solution.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this thing! Here’s how we can approach it:

  1. Label the equations: Let’s call our first equation (3a + 5b + 4c = 90) Equation 1, and the second equation (5a + 3b + 4c = 70) Equation 2. This simple step helps us keep track of what we're doing.
  2. Subtract the equations: This is where the magic happens. Subtract Equation 2 from Equation 1. This means we do: (3a + 5b + 4c) - (5a + 3b + 4c) = 90 - 70. Notice how we're subtracting the entire left side of Equation 2 from the entire left side of Equation 1, and doing the same on the right side. This keeps our equation balanced.
  3. Simplify: After the subtraction, we simplify. The left side becomes: 3a + 5b + 4c - 5a - 3b - 4c. Notice that the 4c terms cancel out (4c - 4c = 0). Combining the a terms (3a - 5a) gives us -2a, and combining the b terms (5b - 3b) gives us 2b. So, the left side simplifies to -2a + 2b. The right side is simply 90 - 70, which equals 20. Our new equation is: -2a + 2b = 20.
  4. Divide by 2: To make things even simpler, we can divide both sides of the equation by 2. This gives us: -a + b = 10. We're getting closer!
  5. Rearrange (Optional): We can rewrite this as b - a = 10. This form might be easier to work with for some people – it’s totally a matter of preference.
  6. Add the original equations: Now, let’s add Equation 1 and Equation 2 together: (3a + 5b + 4c) + (5a + 3b + 4c) = 90 + 70.
  7. Simplify: Combining like terms on the left side, we get 8a + 8b + 8c. On the right side, 90 + 70 equals 160. Our equation now looks like this: 8a + 8b + 8c = 160.
  8. Divide by 8: See a common factor? We can divide both sides of the equation by 8. This simplifies our equation to: a + b + c = 20. Boom! That’s exactly what we wanted to find!

And there you have it! We've successfully determined that a + b + c = 20. The key was to manipulate the equations strategically, rather than trying to solve for each variable individually. This approach highlights the beauty of algebra – finding clever ways to simplify and solve problems.

Alternative Approaches

While our step-by-step solution is pretty slick, it's always good to know there's more than one way to crack an egg, right? In math, alternative approaches can sometimes offer a different perspective or even be more efficient for certain people. So, let's explore a couple of other ways we could have tackled this problem. Understanding different methods not only broadens your mathematical toolkit but also deepens your understanding of the underlying concepts.

Using Substitution (Less Efficient in this case)

One common technique for solving systems of equations is substitution. While it might not be the most efficient route here, let's see how it would work.

  1. Solve for one variable: We could pick one of our original equations and solve for one variable in terms of the others. For instance, from Equation 1 (3a + 5b + 4c = 90), we could solve for a:
    • 3a = 90 - 5b - 4c
    • a = (90 - 5b - 4c) / 3
  2. Substitute: Now, we'd substitute this expression for a into Equation 2 (5a + 3b + 4c = 70). This would give us an equation with just b and c.
  3. Simplify: After substituting, we'd simplify and try to solve for b or c. This would likely involve some algebraic manipulation and might get a bit messy.
  4. Back-substitute: Once we found a value for b or c, we'd back-substitute into our earlier equations to find the other variables and then calculate a + b + c.

As you can see, this method involves quite a few steps and some potentially complicated fractions. It works, but it's definitely more involved than our earlier approach of subtracting and adding equations. This illustrates a key point in problem-solving: choosing the right strategy can make a huge difference in efficiency!

Linear Combination (Similar to our main method)

Another approach, which is actually quite similar to our main method, is called linear combination. This involves multiplying the equations by constants and then adding or subtracting them to eliminate variables.

  1. Target a variable for elimination: We could aim to eliminate c, for example, since it has the same coefficient (4) in both equations. (This is what we implicitly did in our main method when we subtracted the equations.)
  2. Multiply (if necessary): In this case, we don't need to multiply by any constants since the c terms already have the same coefficient.
  3. Add or Subtract: We'd subtract Equation 2 from Equation 1, just like we did before, to eliminate c. This would give us an equation in terms of a and b.
  4. Target another variable: Next, we could try to eliminate either a or b from the original equations (or the new equation we got in step 3 along with one of the originals). This might involve multiplying one or both equations by constants before adding or subtracting.
  5. Solve the resulting system: Eventually, we'd end up with a system of equations that we could solve for a, b, and c individually (if we wanted to), or we could manipulate the equations to directly find a + b + c.

Linear combination is a powerful technique, and it’s essentially the more formalized version of the method we used initially. It's especially useful when dealing with larger systems of equations with more variables.

Key Takeaways

So, what have we learned from this mathematical adventure? Here are some key takeaways:

  • Strategic manipulation is key: In many algebraic problems, the trick isn't just about grinding through calculations, but about finding clever ways to manipulate equations. Look for opportunities to add, subtract, multiply, or divide equations in ways that simplify the problem.
  • Know what you're solving for: In our case, we didn't need to find a, b, and c individually. Recognizing that we only needed a + b + c allowed us to use a more efficient method.
  • Alternative approaches exist: There's often more than one way to solve a math problem. Exploring different methods can deepen your understanding and help you develop your problem-solving skills.
  • Simplify, simplify, simplify: Always look for opportunities to simplify equations. Dividing by common factors, combining like terms, and rearranging expressions can make the problem much more manageable.

Practice Problems

Want to put your newfound skills to the test? Try solving these similar problems:

  1. If 2x + 3y + z = 15 and 4x + y + z = 10, find the value of x + y + z.
  2. Given that p + 2q + 3r = 30 and 2p + q + 3r = 25, determine the value of p + q + r.

Remember, the key is to look for ways to combine the equations strategically. Don't be afraid to experiment and try different approaches!

Conclusion

Alright, guys, we've successfully navigated this math problem and learned some valuable techniques along the way! Solving for a + b + c when given two equations might seem daunting at first, but by using strategic manipulation and understanding the problem's core, we found a pretty elegant solution. Remember, math isn't just about memorizing formulas – it's about problem-solving and creative thinking. So, keep practicing, keep exploring, and keep having fun with math! This journey of solving equations enhances not only our mathematical skills but also our analytical thinking, which is valuable in many aspects of life. Whether you're tackling complex algebraic problems or making everyday decisions, the ability to break down a problem and find the most efficient solution is a skill that will serve you well. Keep challenging yourself, and you'll be amazed at what you can achieve!