Solving The System Of Equations -28x + 24y + 24z = -5, -4x + 24y = -3, -4x + 28y - 12z = -11

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In the realm of mathematics, solving systems of linear equations is a fundamental skill with applications spanning various fields, from engineering and physics to economics and computer science. This article delves into the process of solving a specific system of three linear equations with three unknowns. We will explore techniques such as elimination and substitution to systematically arrive at the solution. Understanding these methods is crucial for anyone seeking to master linear algebra and its practical applications. So, let's embark on this journey of unraveling the solution to the given system of equations.

The System of Equations

We are presented with the following system of linear equations:

  1. -28x + 24y + 24z = -5
  2. -4x + 24y = -3
  3. -4x + 28y - 12z = -11

This system comprises three equations, each containing three variables: x, y, and z. Our goal is to find the values of these variables that simultaneously satisfy all three equations. This means that when we substitute the solutions for x, y, and z into each equation, the equation holds true. Solving systems of linear equations is a cornerstone of linear algebra, and various methods can be employed to find the solution. Let's delve into the step-by-step process of solving this specific system, highlighting the techniques and reasoning involved.

Step 1: Elimination of a Variable

The elimination method is a powerful technique for solving systems of equations. It involves manipulating the equations to eliminate one variable, thereby reducing the system to a simpler form. In this case, we can eliminate x from equations (1) and (3) using equation (2). The key idea behind elimination is to create coefficients for a particular variable that are opposites of each other. When we add the equations, the terms with that variable cancel out, leaving us with an equation in fewer variables.

To eliminate x from equation (1), we can multiply equation (2) by -7. This will give us a 28x term in the modified equation (2), which is the opposite of the -28x term in equation (1). Similarly, to eliminate x from equation (3), we can subtract equation (2) from equation (3). This is because both equations (2) and (3) already have the same coefficient for x, which is -4. By subtracting, we effectively cancel out the x terms. The elimination method is not just a mathematical trick; it's a systematic way of simplifying complex systems of equations. It allows us to reduce the number of variables and equations, making the problem more manageable and leading us closer to the solution. Let's see how this plays out in the next step.

Step 2: Modified Equations

Following the elimination strategy, we perform the following operations:

  • Multiply equation (2) by -7: This yields 28x - 168y = 21.
  • Subtract equation (2) from equation (3): This gives us (-4x + 28y - 12z) - (-4x + 24y) = -11 - (-3), which simplifies to 4y - 12z = -8.

Now, let's add the modified equation (2) to equation (1):

(-28x + 24y + 24z) + (28x - 168y) = -5 + 21. This simplifies to -144y + 24z = 16. This new equation, along with the equation 4y - 12z = -8 obtained earlier, forms a reduced system of two equations with two variables. The significance of these modifications is that we have successfully eliminated the variable x from two of the equations. We have effectively reduced the complexity of the original system. Instead of dealing with three equations and three unknowns, we now have a smaller system involving only y and z. This is a major step towards solving the entire system. By focusing on these two variables, we can apply further techniques to isolate their values. The process of simplification is key to solving complex mathematical problems. By breaking down the problem into smaller, more manageable parts, we can systematically work towards the solution.

Step 3: Further Simplification

We now have a system of two equations:

  1. -144y + 24z = 16
  2. 4y - 12z = -8

To further simplify, we can divide the first equation by 8, resulting in -18y + 3z = 2. We can also divide the second equation by 4, leading to y - 3z = -2. This simplification is crucial because it makes the coefficients smaller and easier to work with. Smaller coefficients reduce the chances of making arithmetic errors, especially when performing further operations such as elimination or substitution. The act of simplifying equations is not just about making them look cleaner; it's about enhancing their manageability and reducing the risk of errors in subsequent calculations. Furthermore, simplified equations often reveal underlying relationships and patterns that might be obscured in their more complex forms. In this case, the simplification has set the stage for the next step, where we can easily eliminate another variable.

Step 4: Eliminating Another Variable

Looking at the simplified equations, we can readily eliminate z. To do this, we can add the equation -18y + 3z = 2 to the multiple of equation y - 3z = -2. Notice that the coefficients of z are already opposites (3 and -3), which makes the elimination straightforward. This strategic move to eliminate z is a critical step in our solution process. By focusing on eliminating variables one at a time, we systematically reduce the complexity of the system. This approach is a hallmark of many algebraic techniques. It allows us to break down a seemingly daunting problem into smaller, more manageable steps. The beauty of this method lies in its ability to isolate variables, leading us closer to the solution. As we proceed with this elimination, we'll see how the value of y emerges, paving the way for finding the values of the remaining variables.

Step 5: Solving for y

Multiplying the equation y - 3z = -2 by -1 gives us -y + 3z = 2. Now, adding this to -18y + 3z = 2, we get:

(-18y + 3z) + (-y + 3z) = 2 + 2. Simplifying this, we get -19y = 4. Therefore, y = -4/19. This step marks a significant breakthrough in our solution. We have successfully isolated the value of one of the variables, y. This is a crucial milestone because once we know the value of one variable, we can substitute it back into other equations to find the values of the remaining variables. The process of solving for y demonstrates the power of systematic elimination and simplification. By carefully manipulating the equations, we were able to create a situation where y could be isolated and its value determined. This is a testament to the elegance and effectiveness of algebraic techniques. Now that we have the value of y, we are well on our way to unraveling the complete solution to the system.

Step 6: Solving for z

Now that we have y = -4/19, we can substitute this value back into the equation y - 3z = -2 to solve for z. This gives us:

(-4/19) - 3z = -2. Adding 4/19 to both sides, we get -3z = -2 + (4/19), which simplifies to -3z = -34/19. Dividing both sides by -3, we find z = 34/57. This step is a classic example of how substitution is used in solving systems of equations. Once we have the value of one variable, we can plug it into another equation to find the value of a second variable. This process is like a domino effect: finding one value leads to the next, and so on. The accuracy of this step is crucial, as any error in calculating z will propagate to the final step when we solve for x. Therefore, careful arithmetic and attention to detail are paramount. By finding the value of z, we have further narrowed down the solution. We now know the values of y and z, leaving only x to be determined. The final piece of the puzzle is within reach, and we are about to complete the solution.

Step 7: Solving for x

Finally, we substitute the values of y = -4/19 into the equation -4x + 24y = -3 to solve for x. This gives us:

-4x + 24(-4/19) = -3. This simplifies to -4x - 96/19 = -3. Adding 96/19 to both sides, we get -4x = -3 + 96/19, which simplifies to -4x = 39/19. Dividing both sides by -4, we find x = -39/76. This final step is the culmination of our efforts. We have successfully determined the value of x, the last unknown variable in the system. The substitution method, once again, has played a key role in this process. By plugging in the known values of y and z, we were able to isolate x and solve for its value. The result, x = -39/76, completes the solution set for the system of equations. It's important to remember that each step in this process builds upon the previous one. Any error along the way would affect the final answer. Therefore, meticulous attention to detail and accuracy are essential when solving systems of equations. With x, y, and z all determined, we have successfully navigated the complexities of the system and arrived at the solution.

The Solution

Therefore, the solution to the system of equations is:

  • x = -39/76
  • y = -4/19
  • z = 34/57

This solution set represents the unique values of x, y, and z that simultaneously satisfy all three equations in the system. To verify this solution, we can substitute these values back into the original equations and check if they hold true. This is a crucial step in the problem-solving process, as it ensures that our solution is accurate and reliable. The process of solving systems of linear equations is not just about finding the numbers; it's about understanding the relationships between the variables and the equations. It's about applying logical reasoning and algebraic techniques to unravel the solution. The solution we have found is not just a set of numbers; it's a testament to the power of mathematical methods and our ability to solve complex problems.

Conclusion

Solving systems of linear equations is a fundamental skill in mathematics with wide-ranging applications. We have demonstrated a systematic approach using elimination and substitution to solve a specific system of three equations. Understanding these techniques is essential for anyone working with mathematical models and real-world problems. The journey of solving this system of equations has highlighted several key principles. We've seen the importance of simplification, elimination, and substitution. We've emphasized the need for accuracy and attention to detail. And we've demonstrated how a complex problem can be broken down into smaller, more manageable steps. These principles are not just applicable to solving systems of equations; they are valuable skills in any problem-solving endeavor. The ability to think systematically, to identify key strategies, and to execute them with precision is a hallmark of mathematical thinking. As we conclude this exploration, we hope that the techniques and insights gained here will empower you to tackle other mathematical challenges with confidence and skill. The world of mathematics is vast and fascinating, and solving systems of equations is just one piece of the puzzle. But it's a crucial piece, and mastering it opens doors to a deeper understanding of the mathematical world.