Solving Systems Of Linear Equations Analytically A Step-by-Step Guide

Solving systems of linear equations is a fundamental concept in mathematics with applications across various fields, including engineering, economics, and computer science. When faced with a system of linear equations, analytical methods provide a precise way to find the solution set. In this article, we'll dive deep into how to solve such systems analytically, using the given system as a practical example. Let's get started, guys!

Understanding the System of Equations

Before we jump into the solution, let's clearly understand the system we're dealing with. We have the following three linear equations:

1. 2x - 4y + 2z = 8
2. 5x - 4y - 5z = -38
3. x - y + 3z = 31

Our main goal here is to find the values of x, y, and z that satisfy all three equations simultaneously. These values, when found, form the solution set of the system. There are several analytical methods to tackle this, such as substitution, elimination, and using matrices. Each method has its own strengths, and the choice often depends on the specific system at hand. For this particular system, we'll use the elimination method because it's pretty straightforward and effective for systems of this size. The elimination method involves manipulating the equations to eliminate one variable at a time, making the system simpler to solve.

When you look at these equations, you might think, "Okay, where do I even start?" Don't worry, we'll break it down step by step. The key is to strategically eliminate variables. For instance, notice that the coefficients of 'y' in the first two equations are -4. This gives us a good starting point for elimination. By subtracting one equation from the other, we can get rid of 'y' and end up with an equation involving only 'x' and 'z'. This is how we simplify the problem into manageable chunks. Remember, the goal is always to reduce the complexity and make the system easier to solve. The beauty of the elimination method lies in its systematic approach, which helps avoid confusion and ensures a clear path towards the solution. As we move through the steps, I'll highlight the reasoning behind each manipulation, so you not only see the process but also understand why it works. This way, you can apply the same principles to solve different systems of equations. Cool, right?

Applying the Elimination Method

The elimination method, as we mentioned, is a powerful technique for solving systems of linear equations. The core idea behind this method is to strategically manipulate the equations to eliminate one variable at a time. This simplification process leads us closer to finding the values of each variable that satisfy the system. It’s like peeling an onion – layer by layer – until you reach the core.

So, how do we apply this to our system? Let’s start by focusing on the first two equations:

1. 2x - 4y + 2z = 8
2. 5x - 4y - 5z = -38

Notice that the coefficients of y are the same (-4) in both equations. This is perfect for elimination! To eliminate y, we can subtract the first equation from the second. When we do this, the y terms will cancel out, giving us a new equation in terms of x and z. This is a key step in reducing the complexity of the system. Think of it as narrowing down the possibilities, like a detective eliminating suspects in a crime case. The new equation will provide valuable information about the relationship between x and z, bringing us one step closer to the solution.

Let's perform the subtraction:

(5x - 4y - 5z) - (2x - 4y + 2z) = -38 - 8

Simplifying this, we get:

3x - 7z = -46

Now we have a new equation with only two variables, x and z. This is progress! But we're not done yet. We need to eliminate another variable to solve for the remaining one. To do this, we'll use a similar strategy, but this time we'll work with a different pair of equations. Our goal is to create another equation in terms of x and z, so we can eventually solve for x or z. This is where things start to get interesting, guys! It's like setting up a chain reaction, where each step leads us closer to the final answer. By systematically eliminating variables, we're essentially breaking down a complex problem into smaller, more manageable parts. This approach not only helps us find the solution but also provides a deeper understanding of the relationships between the variables. So, let's keep going and see what we can eliminate next!

Further Elimination and Solving for Variables

Now that we've eliminated y from the first two equations, we have a new equation:

3x - 7z = -46

To continue our elimination strategy, we need to work with another pair of equations to eliminate y again. This time, let's use the first and third equations:

1. 2x - 4y + 2z = 8
2. x - y + 3z = 31

Notice that the coefficients of y are -4 and -1, respectively. To eliminate y, we need to make these coefficients the same. We can do this by multiplying the third equation by -4:

-4(x - y + 3z) = -4(31)

Which simplifies to:

-4x + 4y - 12z = -124

Now we have:

1. 2x - 4y + 2z = 8
2. -4x + 4y - 12z = -124

The coefficients of y are now opposites, which means we can eliminate y by adding these two equations together. This is like combining forces to cancel out a common enemy, leaving us with a clearer path forward.

Adding the equations, we get:

(2x - 4y + 2z) + (-4x + 4y - 12z) = 8 + (-124)

Simplifying, we have:

-2x - 10z = -116

Now we have another equation in terms of x and z. Great job, guys! We're building a system of two equations with two unknowns, which is a classic scenario in algebra. This new system will allow us to solve for x and z, and then we can easily find y. It's like solving a puzzle, where each piece we find helps us complete the bigger picture.

Our two equations in x and z are:

1. 3x - 7z = -46
2. -2x - 10z = -116

To solve this system, we can use the elimination method again. Let's eliminate x. To do this, we need to make the coefficients of x in both equations multiples of each other. We can multiply the first equation by 2 and the second equation by 3:

1. 2(3x - 7z) = 2(-46) -> 6x - 14z = -92
2. 3(-2x - 10z) = 3(-116) -> -6x - 30z = -348

Now the coefficients of x are 6 and -6. We can eliminate x by adding the equations:

(6x - 14z) + (-6x - 30z) = -92 + (-348)

Simplifying, we get:

-44z = -440

Now we can solve for z:

z = -440 / -44

z = 10

Awesome! We've found the value of z, which is a major breakthrough! Now that we know z, we can substitute it back into one of the equations with x and z to solve for x. It's like retracing our steps to find the missing pieces of the puzzle.

Let's use the equation 3x - 7z = -46:

3x - 7(10) = -46

3x - 70 = -46

3x = 24

x = 8

Excellent! We've found the value of x, which is another key piece of the solution. Now that we know x and z, we can substitute these values into any of the original equations to solve for y. It's like the final act of a play, where everything comes together to reveal the grand finale.

Let's use the first original equation, 2x - 4y + 2z = 8:

2(8) - 4y + 2(10) = 8

16 - 4y + 20 = 8

36 - 4y = 8

-4y = -28

y = 7

Fantastic! We've found the value of y, and now we have all the pieces of the puzzle! We've solved for x, y, and z. It's like completing a marathon, and the finish line is finally in sight.

Presenting the Solution Set

After all our hard work, we've successfully found the values of x, y, and z that satisfy the given system of equations. It's time to present our findings in a clear and concise way. The solution set represents the ordered triple (x, y, z) that makes all three equations true. Think of it as the ultimate answer key to our math problem.

So, what values did we find? Let's recap:

• x = 8
• y = 7
• z = 10

Therefore, the solution set is (8, 7, 10). This is the point in three-dimensional space where all three planes represented by our equations intersect. It's a single, unique solution that satisfies the entire system. We can visualize this as the common meeting point of three flat surfaces in space.

To ensure our solution is correct, we should always verify it by substituting these values back into the original equations. This is like double-checking our work to avoid any mistakes. It's a crucial step in the problem-solving process, giving us confidence in our answer.

Let's substitute the values into each equation:

1. 2x - 4y + 2z = 8

   2(8) - 4(7) + 2(10) = 16 - 28 + 20 = 8 (Correct!)

2. 5x - 4y - 5z = -38

   5(8) - 4(7) - 5(10) = 40 - 28 - 50 = -38 (Correct!)

3. x - y + 3z = 31

   8 - 7 + 3(10) = 8 - 7 + 30 = 31 (Correct!)

Our solution set (8, 7, 10) satisfies all three equations, confirming that it is indeed the correct solution. We've successfully navigated the complex world of linear equations and emerged victorious! Now, let's present our final answer in the requested format.

The solution set is {8, 7, 10}.

Conclusion

Solving systems of linear equations analytically can seem daunting at first, but by using methods like elimination, we can break down the problem into manageable steps. We've journeyed through the world of equations, using our analytical skills to uncover the hidden solution. The key is to be systematic and persistent, and always double-check your work to ensure accuracy. Think of it as a mathematical adventure, where each step brings us closer to the treasure – the solution! By understanding the underlying principles and practicing regularly, you can master this important mathematical skill. So, keep exploring, keep solving, and remember that every problem is an opportunity to learn and grow. This example illustrates the power and elegance of analytical methods in mathematics. It's like using a precise tool to craft a perfect solution. The solution set {8, 7, 10} is a testament to our efforts, a symbol of our problem-solving prowess. We've not only found the answer but also gained a deeper understanding of the relationships between variables and the beauty of mathematical reasoning. So, go forth and conquer other mathematical challenges with confidence and enthusiasm! You've got this!