Solution Types

Overview

A system of linear equations $A\mathbf{x} = \mathbf{b}$ can have no solution, exactly one solution, or infinitely many solutions. Understanding when each occurs is fundamental to linear algebra.

Three Possibilities

Type	Number of Solutions	Visual (2D)
Inconsistent	0	Parallel lines
Consistent, unique	1	Intersecting lines
Consistent, infinite	$\infty$	Same line

Determining Solution Type

After row reducing the augmented matrix $[A \mid \mathbf{b}]$:

No Solution (Inconsistent)

A row of the form $[0 \; 0 \; \cdots \; 0 \mid c]$ where $c \neq 0$.

$\left[\begin{array}{ccc|c} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 5 \end{array}\right] \leftarrow 0 = 5 \text{ (contradiction)}$

Unique Solution

Every column of $A$ has a pivot (full column rank).

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

Infinitely Many Solutions

Consistent, but $A$ has at least one free variable (column without pivot).

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

Testing with Rank

For system $A\mathbf{x} = \mathbf{b}$ where $A$ is $m \times n$:

Condition	Result
$\text{rank}(A) < \text{rank}([A \mid \mathbf{b}])$	No solution
$\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) = n$	Unique solution
$\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) < n$	Infinitely many solutions

Free Variables

When infinitely many solutions exist:

• Free variables: correspond to non-pivot columns
• Basic variables: correspond to pivot columns
• Dimension of solution set = number of free variables

Parametric Solutions

Express basic variables in terms of free variables.

Example

RREF:

$\left[\begin{array}{cccc|c} 1 & 3 & 0 & 2 & 5 \\ 0 & 0 & 1 & 4 & 6 \end{array}\right]$

Variables: $x_1, x_2, x_3, x_4$

• Pivot columns: 1, 3 (basic variables: $x_1$, $x_3$)
• Free columns: 2, 4 (free variables: $x_2$, $x_4$)

Solution:

$x_1 = 5 - 3x_2 - 2x_4$ $x_3 = 6 - 4x_4$ $x_2 = s \text{ (free)}$ $x_4 = t \text{ (free)}$

Vector form:

$\mathbf{x} = \begin{bmatrix} 5 \\ 0 \\ 6 \\ 0 \end{bmatrix} + s\begin{bmatrix} -3 \\ 1 \\ 0 \\ 0 \end{bmatrix} + t\begin{bmatrix} -2 \\ 0 \\ -4 \\ 1 \end{bmatrix}$

Homogeneous Systems

For $A\mathbf{x} = \mathbf{0}$:

• Always consistent ($\mathbf{x} = \mathbf{0}$ is always a solution)
• Either unique solution ($\mathbf{x} = \mathbf{0}$) or infinitely many

When Only Trivial Solution

• $A$ has full column rank ($n$ pivots for $m \times n$ matrix)
• Columns of $A$ are linearly independent

When Non-trivial Solutions Exist

• $A$ does not have full column rank
• $n > \text{rank}(A)$
• For $m < n$ (more variables than equations): always non-trivial solutions

General Solution Structure

For $A\mathbf{x} = \mathbf{b}$ (consistent):

$\mathbf{x} = \mathbf{x}_p + \mathbf{x}_h$

where:

• $\mathbf{x}_p$ is any particular solution to $A\mathbf{x} = \mathbf{b}$
• $\mathbf{x}_h$ is the general solution to $A\mathbf{x} = \mathbf{0}$

The solution set is a translate of the null space.

Examples

Example 1: Unique Solution

$x + y = 5$ $x - y = 1$

RREF:

$\left[\begin{array}{cc|c} 1 & 0 & 3 \\ 0 & 1 & 2 \end{array}\right]$

Solution: $x = 3$, $y = 2$

Example 2: No Solution

$x + y = 5$ $2x + 2y = 8$

RREF:

$\left[\begin{array}{cc|c} 1 & 1 & 5 \\ 0 & 0 & -2 \end{array}\right]$

$0 = -2$: No solution

Example 3: Infinitely Many Solutions

$x + y + z = 3$ $2x + 2y + 2z = 6$

RREF:

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right]$

Free: $y = s$, $z = t$

Solution: $x = 3 - s - t$, $y = s$, $z = t$