Solving System Of Equations A + B = 6, C + D = 4, A + C = 7, B + C = 5

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Hey guys! Today, we’re diving into the fascinating world of solving systems of equations. Specifically, we're going to tackle the system:

  • A + B = 6
  • C + D = 4
  • A + C = 7
  • B + C = 5

This might look a bit intimidating at first glance, but don't worry! We'll break it down step-by-step, so you'll be a pro at solving these in no time. Solving systems of equations is a fundamental skill in mathematics, with applications spanning across various fields such as physics, engineering, economics, and computer science. Understanding how to solve these systems efficiently can open doors to tackling complex real-world problems. This guide aims to provide a comprehensive understanding of the methods used to solve this particular system, while also offering insights into the broader context of systems of equations. Whether you're a student looking to ace your math exams or just someone curious about problem-solving, this guide will equip you with the knowledge and confidence to approach similar challenges.

Understanding Systems of Equations

So, what exactly is a system of equations? Well, it's simply a set of two or more equations that involve the same variables. The goal is to find values for those variables that satisfy all equations simultaneously. In our case, we have four equations with four unknowns (A, B, C, and D). This type of system can be solved using several methods, and we'll explore the most common ones. Before we dive into solving our specific system, let's briefly discuss the general strategies for tackling systems of equations. Common methods include substitution, elimination, and matrix methods. Each method has its own advantages and is suitable for different types of systems. For example, the substitution method involves solving one equation for one variable and substituting that expression into another equation. This reduces the number of variables and equations, making the system easier to solve. The elimination method, on the other hand, involves adding or subtracting equations to eliminate one variable at a time. This method is particularly useful when coefficients of variables are multiples of each other. Understanding these methods and when to apply them is crucial for effectively solving systems of equations. Let's now delve into the specific techniques we'll use to solve our system.

Methods to Solve the System

There are a couple of ways we can approach this. Let's start with the elimination method, which is super handy for this kind of problem. The elimination method is a powerful technique for solving systems of equations. It involves manipulating the equations in the system so that, when added or subtracted, one or more variables are eliminated. This simplifies the system, allowing you to solve for the remaining variables. In this specific problem, we will use a combination of elimination and substitution to find the values of A, B, C, and D. The key is to strategically combine equations to cancel out variables, one at a time. This process might seem complex at first, but with practice, it becomes a straightforward and efficient way to solve systems of equations. The elimination method is particularly effective when dealing with linear equations, where variables are raised to the power of one. This method can be applied to systems with any number of equations and variables, making it a versatile tool in mathematics. Now, let's apply this method to our system.

Elimination Method

  1. Pair up equations: Notice how A and C appear in two equations each? Let’s use that to our advantage. We can pair up the equations as follows:

    • Equation 1: A + B = 6
    • Equation 3: A + C = 7

  2. Subtract Equations: Subtract Equation 1 from Equation 3. This will eliminate 'A':

    • (A + C) - (A + B) = 7 - 6
    • C - B = 1

    Let's call this new equation Equation 5. This step is crucial because it reduces the complexity of the system. By eliminating 'A', we now have an equation that relates C and B, which can be combined with other equations involving these variables. The subtraction process is a fundamental aspect of the elimination method. It allows us to isolate variables and simplify the system. The result, C - B = 1, provides a valuable piece of information that will help us solve for B and C. This equation, along with the original equations, forms a new, simpler system that we can continue to work with. The beauty of this method is its systematic approach, which ensures that we progressively eliminate variables until we can solve for the remaining ones.

  3. Another Pair: Now, let's use another pair of equations:

    • Equation 2: C + D = 4
    • Equation 4: B + C = 5

  4. Subtract Again: Subtract Equation 4 from Equation 2. This will eliminate 'C':

    • (C + D) - (B + C) = 4 - 5
    • D - B = -1

    Let’s call this Equation 6. Similar to the previous step, this subtraction eliminates 'C' and provides us with a relationship between D and B. This equation, D - B = -1, is another key piece of information that we will use to solve the system. The strategic subtraction of equations is at the heart of the elimination method. It allows us to isolate variables and create new equations that simplify the problem. Equation 6, along with Equation 5 (C - B = 1), gives us a system of two equations with three variables (B, C, and D). While we still have more variables than equations, these new equations are much easier to work with than the original set. The next step will involve combining these equations to solve for the remaining variables. This iterative process of eliminating variables and simplifying the system is what makes the elimination method so effective.

  5. New System: Now we have a smaller system:

    • Equation 5: C - B = 1
    • Equation 6: D - B = -1

  6. Use Equation 4: Remember Equation 4 (B + C = 5)? We can use this with Equation 5 to solve for B and C.

  7. Add Equations: Add Equation 5 and Equation 4:

    • (C - B) + (B + C) = 1 + 5
    • 2C = 6
    • C = 3

    Adding equations is another important technique in the elimination method. In this case, adding Equation 5 and Equation 4 eliminates 'B' and allows us to directly solve for 'C'. The result, C = 3, is a significant step forward in solving the system. This value can now be substituted back into other equations to find the values of other variables. The process of adding equations is the counterpart to subtraction in the elimination method. It is used to cancel out variables that have opposite signs or coefficients that can be made opposite through multiplication. This technique, along with subtraction, forms the core of the elimination method, allowing us to systematically reduce the complexity of the system until we can solve for each variable.

  8. Solve for B: Substitute C = 3 into Equation 4:

    • B + 3 = 5
    • B = 2

    Now that we have found the value of C, we can easily find the value of B by substituting C = 3 into Equation 4. This process of substitution is a crucial step in solving systems of equations. Once we have found the value of one variable, we can substitute it into other equations to find the values of the remaining variables. The result, B = 2, is another important piece of the puzzle. We now have the values of B and C, and we can use these to find the values of A and D. This back-substitution process is a systematic way of solving for all the variables in the system. It ensures that we use the information we have already gained to find the remaining unknowns. The combination of elimination and substitution is a powerful strategy for solving systems of equations, and this example demonstrates how effectively these techniques can be applied.

  9. Solve for A: Substitute B = 2 into Equation 1:

    • A + 2 = 6
    • A = 4

  10. Solve for D: Substitute C = 3 into Equation 2:

    • 3 + D = 4
    • D = 1

Solution

So, we've got it! The solution to the system of equations is:

  • A = 4
  • B = 2
  • C = 3
  • D = 1

This was a classic example of how the elimination method can be used to solve a system of linear equations. We systematically eliminated variables by adding or subtracting equations, and then used substitution to find the final values. This step marks the culmination of our efforts to solve the system of equations. We have successfully found the values of A, B, C, and D that satisfy all four equations simultaneously. The solution, A = 4, B = 2, C = 3, and D = 1, is the unique set of values that makes all the equations true. This final step is crucial because it confirms that we have indeed found the correct solution. The process of verifying the solution by substituting the values back into the original equations is a good practice to ensure accuracy. The successful solution of this system demonstrates the power and effectiveness of the elimination and substitution methods. These techniques can be applied to a wide range of systems of equations, making them valuable tools in mathematics and other fields.

Verification

To be absolutely sure, let's plug these values back into the original equations and see if they hold true:

  • A + B = 4 + 2 = 6 (Correct!)
  • C + D = 3 + 1 = 4 (Correct!)
  • A + C = 4 + 3 = 7 (Correct!)
  • B + C = 2 + 3 = 5 (Correct!)

Yay! They all check out. This step is a crucial part of the problem-solving process. Verifying the solution ensures that we have not made any errors in our calculations. By substituting the values we found back into the original equations, we can confirm that they satisfy all the equations simultaneously. This verification step provides a high level of confidence in the correctness of our solution. It also reinforces our understanding of the system of equations and how the values of the variables relate to each other. The process of verification is a fundamental aspect of mathematical problem-solving, and it should always be performed to ensure accuracy. In this case, the verification confirms that our solution is indeed correct, and we can confidently say that we have solved the system of equations.

Tips and Tricks

Here are a few extra tips for solving systems of equations:

  • Look for easy eliminations: Sometimes, the equations are set up perfectly for quick eliminations.
  • Multiply equations: If needed, you can multiply one or more equations by a constant to make the coefficients match for elimination.
  • Substitution: If one equation is easily solved for a variable, substitution might be a better approach.

These tips and tricks can significantly enhance your problem-solving skills when dealing with systems of equations. The first tip, looking for easy eliminations, emphasizes the importance of observation and strategic thinking. Sometimes, the equations are arranged in such a way that a simple addition or subtraction can eliminate a variable immediately. Recognizing these situations can save time and effort. The second tip, multiplying equations, highlights the flexibility of the elimination method. By multiplying one or more equations by a constant, we can manipulate the coefficients to create opportunities for elimination. This technique is particularly useful when the coefficients are not already multiples of each other. The third tip, considering substitution, reminds us that there are often multiple approaches to solving a system of equations. If one equation can be easily solved for a variable, substitution might be a more efficient method than elimination. Understanding when to use each method is a key aspect of mastering systems of equations. By incorporating these tips and tricks into your problem-solving toolkit, you will be well-equipped to tackle a wide range of systems of equations.

Conclusion

Solving systems of equations might seem tough at first, but with practice, you'll get the hang of it. The elimination method is a powerful tool, and by breaking the problem down into steps, you can conquer even the most complex systems. Remember to always verify your solution to ensure accuracy. And that’s a wrap, guys! You've now got a solid understanding of how to solve systems of equations using the elimination method. Keep practicing, and you'll become a math whiz in no time! Solving systems of equations is a fundamental skill in mathematics, and mastering this skill can open doors to more advanced topics. The elimination method, as we have seen, is a versatile and effective technique for solving these systems. By breaking down the problem into manageable steps, we can systematically eliminate variables and solve for the unknowns. The key to success is practice and a clear understanding of the underlying principles. Remember to always verify your solutions to ensure accuracy and build confidence in your problem-solving abilities. The journey of learning mathematics is a continuous process, and each problem solved is a step forward. With dedication and perseverance, you can conquer any mathematical challenge. So, keep practicing, keep exploring, and keep growing!