13. Gaussian Elimination for Systems of Linear Equations

Gaussian elimination fundamentally simplifies a system of linear equations using equivalent transformations—operations that preserve the solution set while progressively revealing its structure.

At each step, I verify whether the row operations performed are indeed equivalent transformations—and then interpret the resulting row-echelon form to determine whether the system has no solution, a unique solution, or infinitely many solutions.

In the previous article, we introduced the definition and fundamental properties of linear equations. Today, we focus on a cornerstone technique for solving systems of linear equations: Gaussian elimination. This method holds a central place in linear algebra theory—and is equally indispensable in real-world applications across science and engineering.

What Is Gaussian Elimination?

Gaussian elimination is a systematic procedure that uses elementary row operations to transform a system of linear equations into a simpler, more structured form—typically an upper-triangular or row-echelon matrix—making it straightforward to solve for the unknowns via back substitution.

Steps of Gaussian Elimination

1. Construct the Augmented Matrix: Convert the system of linear equations into its augmented matrix representation.

   For the system:

   $$\begin{align*} a_1x + b_1y + c_1z &= d_1 \\ a_2x + b_2y + c_2z &= d_2 \\ a_3x + b_3y + c_3z &= d_3 \end{align*}$$

   the corresponding augmented matrix is:

   $$\left[\begin{array}{ccc|c} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \end{array}\right]$$

2. Perform Row Operations: Apply elementary row operations to convert the matrix into upper-triangular (or row-echelon) form. These operations include:

   • Swapping two rows,
   • Multiplying a row by a nonzero scalar,
   • Adding a scalar multiple of one row to another row.

3. Back Substitution: Once the matrix is in upper-triangular form, solve for the variables starting from the bottom row and working upward.

Worked Example

Consider the following linear system:

$$\begin{align*} 2x + 3y + z &= 1 \quad \text{(1)} \\ 4x + y - 2z &= -2 \quad \text{(2)} \\ -2x + 5y + 3z &= 7 \quad \text{(3)} \end{align*}$$

Step 1: Construct the Augmented Matrix

The augmented matrix is:

$$\left[\begin{array}{ccc|c} 2 & 3 & 1 & 1 \\ 4 & 1 & -2 & -2 \\ -2 & 5 & 3 & 7 \end{array}\right]$$

When learning Gaussian elimination, first identify the augmented matrix, pivot positions, row operations, row-echelon form, back substitution, and how to diagnose inconsistent or underdetermined systems.

Step 2: Perform Row Operations

• Replace Row 2 with $R_2 - 2R_1$:

  $$R_2 = R_2 - 2R_1$$ $$\left[\begin{array}{ccc|c} 2 & 3 & 1 & 1 \\ 0 & -5 & -4 & -4 \\ -2 & 5 & 3 & 7 \end{array}\right]$$

• Replace Row 3 with $R_3 + R_1$:

  $$R_3 = R_3 + R_1$$ $$\left[\begin{array}{ccc|c} 2 & 3 & 1 & 1 \\ 0 & -5 & -4 & -4 \\ 0 & 8 & 4 & 8 \end{array}\right]$$

• Scale Row 2 by $-\frac{1}{5}$:

  $$\left[\begin{array}{ccc|c} 2 & 3 & 1 & 1 \\ 0 & 1 & \frac{4}{5} & \frac{4}{5} \\ 0 & 8 & 4 & 8 \end{array}\right]$$

• Replace Row 3 with $R_3 - 8R_2$:

  $$R_3 = R_3 - 8R_2$$ $$\left[\begin{array}{ccc|c} 2 & 3 & 1 & 1 \\ 0 & 1 & \frac{4}{5} & \frac{4}{5} \\ 0 & 0 & -\frac{4}{5} & 0 \end{array}\right]$$

Step 3: Back Substitution

Start from the last row:

$$-\frac{4}{5}z = 0 \quad \Rightarrow \quad z = 0$$

Substitute into Row 2:

$$y + \frac{4}{5}z = \frac{4}{5} \quad \Rightarrow \quad y + 0 = \frac{4}{5} \quad \Rightarrow \quad y = \frac{4}{5}$$

Substitute into Row 1:

$$2x + 3\left(\frac{4}{5}\right) + 0 = 1 \quad \Rightarrow \quad 2x + \frac{12}{5} = 1 \quad \Rightarrow \quad 2x = 1 - \frac{12}{5} = -\frac{7}{5} \quad \Rightarrow \quad x = -\frac{7}{10}$$

Thus, the unique solution is:

$$\begin{align*} x &= -\frac{7}{10}, \\ y &= \frac{4}{5}, \\ z &= 0. \end{align*}$$

Before reading “Systems of Linear Equations: Gaussian Elimination”, use the accompanying diagram to confirm the conceptual flow; after reading, revisit which steps you can execute directly—and which require supplementary study.

When reviewing “Systems of Linear Equations: Gaussian Elimination”, place key concepts, procedural steps, and observable outcomes side-by-side on a single page for efficient consolidation.

When practicing “Systems of Linear Equations: Gaussian Elimination”, record input conditions, transformation actions, and resulting outputs together—so they’re readily verifiable next time.

Summary

Gaussian elimination is a robust, systematic method for solving linear systems. By constructing an augmented matrix and applying carefully chosen elementary row operations, we reduce the system to a tractable form—enabling clear analysis and solution. In practice, this technique underpins countless applications in physics, computer science, economics, and engineering.

In the next article, we’ll explore the distinction between homogeneous and nonhomogeneous linear systems—and their respective roles and interpretations. Stay tuned!