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Parametric Solutions
& Free Variables

Learn how free variables give rise to parametric descriptions of solution sets, and understand the geometry behind infinitely many solutions.

Linear Algebra

What Is a System of Linear Equations?

A system of linear equations is a collection of one or more equations involving the same set of variables. We write it as \( A\mathbf{x} = \mathbf{b} \), where \( A \) is a coefficient matrix, \( \mathbf{x} \) is the vector of unknowns, and \( \mathbf{b} \) is the constant vector.

Unique Solution
Exactly one value for each variable satisfies every equation.
Infinitely Many
A family of solutions described by one or more parameters.
No Solution
The equations are inconsistent: no values satisfy all of them.

Pivot vs Free Variables

After row reducing the augmented matrix to echelon form, each column either contains a pivot (leading 1) or it does not.

Pivot Variable
Corresponds to a column with a leading 1. Its value is determined by the other variables.
Free Variable
Corresponds to a column with no pivot. It can take any real value and acts as a parameter.

Reading Pivots from Echelon Form

1 3 -2 5 4 0 0 1 -1 2 0 0 0 0 0 pivot free pivot free \(x_1\) \(x_2\) \(x_3\) \(x_4\)

What Does Parametric Mean?

A parametric solution expresses every variable in terms of one or more free parameters. Each free variable becomes a parameter (often called \( s, t, u, \dots \)) that can take any real value.

$$ \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 8 - 3s - 3t \\ s \\ 2 + t \\ t \end{pmatrix} $$

Vector Form of the Solution

We can split the parametric solution into a particular solution plus a linear combination of direction vectors:

$$ \mathbf{x} = \underbrace{\begin{pmatrix} 8 \\ 0 \\ 2 \\ 0 \end{pmatrix}}_{\text{particular}} + s\underbrace{\begin{pmatrix} -3 \\ 1 \\ 0 \\ 0 \end{pmatrix}}_{\text{direction 1}} + t\underbrace{\begin{pmatrix} -3 \\ 0 \\ 1 \\ 1 \end{pmatrix}}_{\text{direction 2}} $$

Geometric Interpretation

\(x_1\) \(x_2\) \(x_3\) p \( v_1 \) (free s) \( v_2 \) (free t) 0 solution plane
0 free variables
A single point (unique solution)
1 free variable
A line through \( \mathbf{p} \)
2 free variables
A plane through \( \mathbf{p} \)

Homogeneous vs Non-homogeneous

\( A\mathbf{x} = \mathbf{0} \) \( A\mathbf{x} = \mathbf{b} \)
Name Homogeneous Non-homogeneous
Trivial solution Always: \( \mathbf{x} = \mathbf{0} \) Not guaranteed
Solution set shape Subspace through origin Shifted subspace (coset)
Parametric form \( \mathbf{x} = s\mathbf{v}_1 + t\mathbf{v}_2 + \cdots \) \( \mathbf{x} = \mathbf{p} + s\mathbf{v}_1 + t\mathbf{v}_2 + \cdots \)
Free vars \( > 0 \)? Infinitely many solutions Infinitely many or none

The Existence & Uniqueness Theorem

This theorem tells us everything we need: row reduce, count pivots, and read off the answer type.

How to Find the Parametric Solution

1
Write the augmented matrix \( [A \mid \mathbf{b}] \) for the system.
2
Row reduce to reduced row echelon form (RREF).
3
Identify pivot columns and free columns.
4
Assign a parameter (\( s, t, \ldots \)) to each free variable.
5
Express pivot variables in terms of the free parameters using back-substitution from RREF.
6
Write in vector form: \( \mathbf{x} = \mathbf{p} + s\mathbf{v}_1 + t\mathbf{v}_2 + \cdots \)

Example 1: One Free Variable

Solve the system: \( x_1 + 2x_2 = 5 \)

1
Augmented matrix: \(\left[\begin{array}{cc|c} 1 & 2 & 5 \end{array}\right]\). Already in RREF.
2
Column 1 has a pivot, column 2 does not. So \( x_1 \) is a pivot variable and \( x_2 \) is free.
3
Let \( x_2 = t \). Then \( x_1 = 5 - 2t \).
4
Vector form: \( \mathbf{x} = \begin{pmatrix} 5 \\ 0 \end{pmatrix} + t\begin{pmatrix} -2 \\ 1 \end{pmatrix} \).

Example 2: Two Free Variables

Solve: \( x_1 + 3x_2 - 2x_3 + 5x_4 = 4 \) and \( x_3 - x_4 = 2 \).

1
RREF: \(\left[\begin{array}{cccc|c} 1 & 3 & 0 & 3 & 8 \\ 0 & 0 & 1 & -1 & 2 \end{array}\right]\). Pivots in columns 1, 3.
2
Free variables: \( x_2 = s \), \( x_4 = t \).
3
Back-substitute: \( x_3 = 2 + t \), \( x_1 = 8 - 3s - 3t \).
$$ \mathbf{x} = \begin{pmatrix} 8\\0\\2\\0 \end{pmatrix} + s\begin{pmatrix} -3\\1\\0\\0 \end{pmatrix} + t\begin{pmatrix} -3\\0\\1\\1 \end{pmatrix} $$

Example 3: Homogeneous System

Find all solutions to \( A\mathbf{x} = \mathbf{0} \), where \( A = \begin{bmatrix} 1 & 2 & -1 \\ 2 & 4 & -2 \end{bmatrix} \).

1
RREF of the augmented matrix: \(\left[\begin{array}{ccc|c} 1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\). Only 1 pivot (column 1).
2
\( n - r = 3 - 1 = 2 \) free variables: \( x_2 = s \), \( x_3 = t \).
3
\( x_1 = -2s + t \). Solution: \( \mathbf{x} = s\begin{pmatrix} -2\\1\\0 \end{pmatrix} + t\begin{pmatrix} 1\\0\\1 \end{pmatrix} \).

Why This Matters

Null Space
Free variables generate the null space of \( A \): the set of all \( \mathbf{x} \) with \( A\mathbf{x} = \mathbf{0} \).
Rank
The number of pivots is the rank of the matrix. \( \text{Rank} + \text{nullity} = \text{number of columns} \).
Applications
From network flow to curve fitting, parametric solutions describe degrees of freedom in real systems.

Common Mistakes

✕ Wrong column for pivots
Confusing the augmented column with a variable column. The augmented column (right of the bar) is never a variable.
✕ Forgetting a free variable
Every non-pivot column (excluding augmented) yields a free variable. Missing one means an incomplete solution.
✕ Setting free vars to 0
Free variables can be any real number. Setting them all to 0 gives only the particular solution, not the full family.

Summary Table

Scenario Pivots vs Unknowns Solution Set Free Variables
Unique solution \( r = n \) Single point 0
Infinitely many \( r < n \), consistent Line, plane, etc. \( n - r \)
No solution Pivot in augmented col Empty set N/A

Question 1 of 5

A system with 5 unknowns row-reduces to a matrix with 3 pivot columns. If the system is consistent, how many free variables does it have?

Question 2 of 5

The RREF of an augmented matrix is \( \left[\begin{array}{ccc|c} 1 & 0 & 3 & 7 \\ 0 & 1 & -2 & 4 \end{array}\right] \). Which variable is free?

Question 3 of 5

For the system \( x_1 + 2x_2 = 6 \), which parametric vector form is correct (with \( x_2 = t \))?

Question 4 of 5

A homogeneous system \( A\mathbf{x} = \mathbf{0} \) has 4 unknowns and 2 pivot columns. The solution set is:

Question 5 of 5

An augmented matrix row-reduces to \( \left[\begin{array}{cc|c} 1 & 0 & 3 \\ 0 & 0 & 5 \end{array}\right] \). What can you conclude?