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

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In the realm of linear algebra, solving systems of linear equations is a fundamental skill. These systems arise in various applications, from engineering and physics to economics and computer science. In this comprehensive guide, we will delve into the process of solving a specific system of linear equations, providing a step-by-step approach and highlighting the underlying concepts.

Understanding the System

At its core, the system of equations we're tackling is represented in matrix form as follows:

\begin{bmatrix} 2 & 0 & -1 \\ -6 & 0 & 3 \\ -2 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}

This translates into a set of three linear equations:

  1. 2*x₁ - x₃ = 0
  2. -6x₁ + 3x₃ = 0
  3. -2*x₁ + x₃ = 0

Our goal is to find the values of x₁, x₂, and x₃ that satisfy all three equations simultaneously. This system is homogeneous because the right-hand side of each equation is zero. Homogeneous systems always have at least one solution, the trivial solution where all variables are zero. However, we're interested in finding out if there are any non-trivial solutions as well.

Gaussian Elimination: A Powerful Technique

One of the most effective methods for solving systems of linear equations is Gaussian elimination. This technique involves transforming the original matrix into an echelon form through a series of elementary row operations. These operations include:

  • Swapping two rows
  • Multiplying a row by a non-zero constant
  • Adding a multiple of one row to another row

The augmented matrix for our system is:

\begin{bmatrix} 2 & 0 & -1 & 0 \\ -6 & 0 & 3 & 0 \\ -2 & 0 & 1 & 0 \end{bmatrix}

Let's perform Gaussian elimination:

  1. Eliminate the -6 in the second row: Add 3 times the first row to the second row:

    \begin{bmatrix} 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ -2 & 0 & 1 & 0 \end{bmatrix}

  2. Eliminate the -2 in the third row: Add the first row to the third row:

    \begin{bmatrix} 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}

Now, the matrix is in echelon form. We can see that the second and third rows are all zeros, indicating that we have fewer independent equations than variables. This implies that there are infinitely many solutions.

Interpreting the Echelon Form

The echelon form of the matrix corresponds to the following system of equations:

  1. 2*x₁ - x₃ = 0

Notice that we only have one equation now. We can express x₁ in terms of x₃: x₁ = (1/2)*x₃. x₂ is a free variable, meaning it can take any value. Let's introduce a parameter t for x₃ and a parameter s for x₂:

  • x₃ = t
  • x₂ = s

Then, x₁ = (1/2)*t.

General Solution: A Family of Solutions

The general solution to the system can be written in vector form:

\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = s \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + t \begin{bmatrix} 1/2 \\ 0 \\ 1 \end{bmatrix}

Where s and t are any real numbers. This means that there are infinitely many solutions, each corresponding to a different choice of s and t. The solution is a linear combination of two vectors, which span the solution space.

Implications of Infinite Solutions

Having infinite solutions in a system of linear equations indicates that the equations are not independent. In this particular case, the second and third equations are multiples of the first equation. This means they don't provide any new information about the relationships between the variables.

The Role of the Free Variable

The presence of the free variable x₂ is crucial in understanding the solution space. It signifies that there's a degree of freedom in choosing the values of the variables. For each value we assign to x₂, we get a different solution to the system.

Applications in Real-World Scenarios

Systems with infinitely many solutions arise in various practical situations. For example, in network analysis, if there are more unknowns than independent equations, the system will have infinitely many solutions. Similarly, in economic modeling, there might be multiple equilibria that satisfy a given set of conditions.

Verification and Examples

To verify our solution, we can substitute the general solution back into the original equations. Let's take an example. Suppose s = 1 and t = 2. Then:

  • x₁ = (1/2)*2 = 1
  • x₂ = 1
  • x₃ = 2

Substituting these values into the original equations:

  1. 2*(1) - 2 = 0 (True)
  2. -6*(1) + 3*(2) = 0 (True)
  3. -2*(1) + 2 = 0 (True)

The solution satisfies all three equations. We can try different values of s and t to generate other solutions and verify that they also work.

Illustrative Examples with Varying Parameters

Let's explore some concrete examples by varying the parameters s and t:

  • Case 1: s = 0, t = 0

    • x₁ = (1/2) * 0 = 0
    • x₂ = 0
    • x₃ = 0 This gives us the trivial solution (0, 0, 0).

  • Case 2: s = 1, t = 0

    • x₁ = (1/2) * 0 = 0
    • x₂ = 1
    • x₃ = 0 This yields the solution (0, 1, 0).

  • Case 3: s = 0, t = 2

    • x₁ = (1/2) * 2 = 1
    • x₂ = 0
    • x₃ = 2 This results in the solution (1, 0, 2).

  • Case 4: s = -1, t = 4

    • x₁ = (1/2) * 4 = 2
    • x₂ = -1
    • x₃ = 4 This gives us the solution (2, -1, 4).

These examples underscore the fact that we can generate an infinite number of solutions by choosing different values for the parameters s and t. Each combination provides a valid solution to the system of linear equations.

Conclusion: Mastering Linear Systems

Solving systems of linear equations is a cornerstone of linear algebra. In this guide, we've systematically tackled a homogeneous system, demonstrating the power of Gaussian elimination and the interpretation of echelon forms. We've seen that systems can have unique solutions, no solutions, or infinitely many solutions. In the case of infinitely many solutions, the general solution provides a complete description of the solution space.

Understanding the concepts and techniques presented here will equip you with the necessary tools to solve a wide range of linear systems encountered in mathematics, science, and engineering. By practicing and applying these methods, you'll gain confidence in your ability to analyze and solve linear systems effectively.

Further Exploration

To deepen your understanding, consider exploring the following topics:

  • Rank of a matrix: The rank of a matrix is the number of linearly independent rows or columns. It provides information about the number of independent equations in a system.
  • Null space of a matrix: The null space is the set of all solutions to the homogeneous equation Ax = 0. It's a vector space that provides insight into the structure of the solutions.
  • Linear independence: Understanding linear independence is crucial for determining whether a system has a unique solution or infinitely many solutions.
  • Applications of linear systems: Explore real-world applications of linear systems in fields like circuit analysis, structural engineering, and computer graphics.

By delving into these areas, you'll expand your knowledge of linear algebra and enhance your problem-solving skills.