Degrees of Freedom Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
The system {x+y+z=6x+y+z=62x−y=1\begin{cases} x + y + z = 6 \\ x + y + z = 6 \\ 2x - y = 1 \end{cases} has 3 equations and 3 variables. Does it have 0 degrees of freedom?

Solution

  1. 1
    Step 1: Equations 1 and 2 are identical (redundant). Only 2 independent equations.
  2. 2
    Step 2: DOF=3−2=1\text{DOF} = 3 - 2 = 1, not 0.
  3. 3
    Step 3: One variable is free; infinitely many solutions.

Answer

11 degree of freedom (not 0)
The count uses independent equations, not total equations. Redundant equations don't reduce degrees of freedom. Always check for linear dependence among the equations.

About Degrees of Freedom

The number of independent values that remain free to be chosen after all constraints in a system have been satisfied.

Learn more about Degrees of Freedom →

More Degrees of Freedom Examples

Example 1 easy

A system has 3 variables and 2 independent equations. How many degrees of freedom?

Example 3 easy

How many DOF does a single equation [formula] have?

Example 4 medium

A system of 4 equations in 4 unknowns has rank 3. How many DOF?

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