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

Hey guys! Today, we're diving into the fascinating world of solving systems of linear equations. It might sound intimidating, but trust me, it's like piecing together a puzzle. We'll tackle a specific example, but the principles we learn will apply to any similar problem you encounter. So, buckle up and let's get started!

Understanding Systems of Linear Equations

Before we jump into the nitty-gritty, let's make sure we're all on the same page. Systems of linear equations are sets of two or more linear equations containing the same variables. A linear equation is simply an equation where the highest power of any variable is 1. Think of it as a straight line when graphed.

The solution to a system of linear equations is a set of values for the variables that makes all the equations true simultaneously. Geometrically, this corresponds to the point(s) where the lines (or planes in 3D) intersect. A system can have one solution, infinitely many solutions, or no solution at all.

In our case, we have a system of three linear equations with three variables (x, y, and z):

y + z = -2
-2x + y - z = 6
5x + y + 6z = 10

Our mission, should we choose to accept it (and we do!), is to find the values of x, y, and z that satisfy all three equations. We'll explore a powerful technique called Gaussian elimination to crack this code. So, let’s dive deep into the methods for tackling these equations and ensuring we find the correct solution, or understand why one doesn’t exist. This process is not just about finding numbers; it’s about understanding the relationships between the equations themselves. We’ll break down each step, ensuring that you not only see the solution but also grasp the underlying concepts. Linear equations are fundamental in various fields, from engineering to economics, making this a crucial skill to master. Understanding how to solve these systems opens doors to solving more complex problems and modeling real-world scenarios with accuracy and confidence. Let’s embark on this journey together and unlock the power of linear equations!

Gaussian Elimination: Our Super Tool

Gaussian elimination is a systematic method for solving systems of linear equations. It involves transforming the system into an equivalent system that's easier to solve. The key idea is to use elementary row operations to create zeros in strategic locations, effectively isolating the variables. These operations include:

1. Swapping two equations (rows).
2. Multiplying an equation by a non-zero constant.
3. Adding a multiple of one equation to another.

Our goal is to transform the system into row-echelon form. This means the first non-zero entry in each row (called the pivot) is to the right of the pivot in the row above it, and all entries below a pivot are zero. Once we're in row-echelon form, we can easily solve for the variables using back-substitution.

Step-by-Step Solution

Let's apply Gaussian elimination to our system:

y + z = -2  (Equation 1)
-2x + y - z = 6  (Equation 2)
5x + y + 6z = 10 (Equation 3)

1. Swapping Equations:

As suggested, let's swap Equation 1 and Equation 2 to get a 'x' term in the first equation. This makes it easier to eliminate 'x' from the subsequent equations.

-2x + y - z = 6  (New Equation 1)
y + z = -2  (New Equation 2)
5x + y + 6z = 10 (Equation 3)

Swapping equations is a crucial step in Gaussian elimination. It allows us to rearrange the system to make the elimination process smoother. By placing equations with leading variables at the top, we can systematically eliminate variables from the equations below. This rearrangement doesn't change the solution set of the system; it merely organizes the equations in a more convenient way for solving. This step highlights the flexibility we have in manipulating systems of equations. The order in which the equations are presented doesn't dictate the solution; we can rearrange them to suit our needs. This is a powerful tool that can simplify complex systems and make them more manageable.

2. Eliminating 'x' from Equation 3:

To eliminate 'x' from Equation 3, we'll add 5/2 times Equation 1 to Equation 3:

Equation 3 + (5/2) * Equation 1:
(5x + y + 6z) + (5/2)(-2x + y - z) = 10 + (5/2)(6)
5x + y + 6z - 5x + (5/2)y - (5/2)z = 10 + 15
(7/2)y + (7/2)z = 25

So our new system is:

-2x + y - z = 6
y + z = -2
(7/2)y + (7/2)z = 25

Eliminating variables is the heart of Gaussian elimination. By strategically combining equations, we can create zeros in specific locations, simplifying the system step by step. This process relies on the principle that adding a multiple of one equation to another doesn't change the solution set. It's like adding the same thing to both sides of an equation – it maintains the balance. This step is crucial for transforming the system into a form that's easier to solve, ultimately leading us to the values of x, y, and z. The careful selection of multipliers ensures that we eliminate the desired variable without disrupting the rest of the system. It’s a delicate dance of numbers, but with each successful elimination, we get closer to the final solution.

3. Analyzing the System:

Notice that Equation 3 can be simplified by multiplying both sides by 2/7:

y + z = 50/7

Now we have:

-2x + y - z = 6
y + z = -2
y + z = 50/7

Observe that the second and third equations, y + z = -2 and y + z = 50/7, are inconsistent. They cannot both be true at the same time. This indicates that the system has no solution. The lines (or planes in 3D) represented by these equations do not intersect at any point. Identifying inconsistencies is a crucial skill in solving systems of equations. It saves us time and effort by preventing us from chasing a solution that doesn't exist. These inconsistencies often arise from conflicting information within the equations themselves. In this case, the two equations involving 'y' and 'z' contradict each other, signaling that no set of values can satisfy all the equations simultaneously. Recognizing these contradictions is a key part of the problem-solving process.

Final Answer:

$$(x, y, z) otin R ^3$$

Key Takeaways

• Gaussian elimination is a powerful technique for solving systems of linear equations.
• Elementary row operations are the tools we use to manipulate the equations.
• Row-echelon form makes it easy to solve for the variables using back-substitution.
• Inconsistent equations indicate that the system has no solution.

Why is This Important?

Solving systems of linear equations is a fundamental skill in mathematics and has wide-ranging applications in various fields, including:

• Engineering: Solving for currents in electrical circuits, analyzing structural stability.
• Computer Graphics: Transformations, projections, and 3D modeling.
• Economics: Modeling supply and demand, optimizing resource allocation.
• Data Science: Linear regression, machine learning algorithms.

By mastering these techniques, you're equipping yourself with a powerful tool for solving real-world problems.

Let's Practice!

To solidify your understanding, try solving the following system of equations using Gaussian elimination:

x - y + z = 2
2x + y - z = 1
3x - y + 2z = 5

Remember, practice makes perfect! The more you work with these concepts, the more comfortable and confident you'll become. So, grab a pencil and paper, and give it a shot!

Solving systems of linear equations is not just an academic exercise; it’s a practical skill that can be applied in countless situations. Whether you're designing a bridge, predicting market trends, or developing a new algorithm, the ability to solve these systems will be invaluable. The key is to approach each problem systematically, break it down into manageable steps, and don't be afraid to experiment. And most importantly, remember that even if you encounter inconsistencies or no solutions, that's still a valuable outcome. It means you've gained a deeper understanding of the system and the relationships between the equations. So keep practicing, keep exploring, and keep unlocking the power of linear equations!