Degrees of Freedom Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Degrees of Freedom.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

If x+y=10x + y = 10, you can choose xx freely, but then yy is fixed. One degree of freedom.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Degrees of freedom count how many values you can still choose freely after every constraint is imposed.

Common stuck point: The procedure for degrees of freedom is the easy part; the trap is counting dependent equations as constraints. Asking "After applying all independent constraints, how many values can I still choose freely?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: After applying all independent constraints, how many values can I still choose freely?

Worked Examples

Example 1

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

Answer

11 degree of freedom

First step

1
Step 1: Apply DOF=n−r\text{DOF} = n - r where n=3n = 3 variables, r=2r = 2 equations.

Full solution

  1. 2
    Step 2: DOF=3−2=1\text{DOF} = 3 - 2 = 1.
  2. 3
    This means the solution is a line (one free parameter).
Degrees of freedom tells you the dimension of the solution space. With 1 DOF, you can freely choose one variable and the others are determined — the solutions form a line in 3D space.

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?

Example 3

medium
A linear system in 55 unknowns has matrix rank 33. How many degrees of freedom in the solution set?

Example 4

medium
A system has 33 equations in 55 unknowns; one equation is the sum of the other two. How many degrees of freedom?

Example 5

hard
A linear system has augmented matrix that row-reduces to two pivot rows in 44 unknowns and is consistent. How many degrees of freedom and how many solutions?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
How many DOF does a single equation x+y=5x + y = 5 have?

Example 2

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

Example 3

easy
How many degrees of freedom does x+y=10x + y = 10 have (two unknowns)?

Example 4

easy
Three independent equations constrain three unknowns. How many degrees of freedom remain?

Example 5

easy
A single point in the plane has how many degrees of freedom unconstrained?

Example 6

easy
After fixing x=3x = 3, how many degrees of freedom does the pair (x,y)(x,y) have?

Example 7

easy
Two unknowns, two independent equations. Degrees of freedom?

Example 8

easy
A line in the plane: how many degrees of freedom does a point ON it have?

Example 9

easy
How many independent equations are needed to fix 44 unknowns uniquely?

Example 10

easy
Does an inequality constraint like x≥0x \ge 0 reduce the COUNT of free variables?

Example 11

medium
System: x+y=4x+y=4 and 2x+2y=82x+2y=8. How many degrees of freedom remain?

Example 12

medium
Three equations in three unknowns, but one is the sum of the other two. Degrees of freedom?

Example 13

medium
A triangle is defined by 33 vertices in the plane. How many degrees of freedom?

Example 14

medium
A system has 55 unknowns and 33 independent equations. How many free parameters in the solution?

Example 15

medium
How many degrees of freedom does a circle in the plane have (center + radius)?

Example 16

medium
If a system has 00 degrees of freedom but contains a contradiction, how many solutions does it have?

Example 17

medium
A point must lie on BOTH a given line and a given circle. Generic degrees of freedom of that point?

Example 18

challenge
A system of 44 equations in 44 unknowns has exactly one redundant equation and is consistent. How many solutions?

Example 19

challenge
Parametrize the solution of x+2y+z=0x + 2y + z = 0 and count degrees of freedom.

Example 20

challenge
A rigid rod in the plane (length fixed). How many degrees of freedom?

Example 21

medium
A plane in 3D space (ax+by+cz=dax+by+cz=d): how many degrees of freedom does a point ON it have?

Example 22

medium
A system of 22 independent equations in 44 unknowns: degrees of freedom?

Example 23

easy
A system has 66 variables and 44 independent equations. How many degrees of freedom?

Example 24

easy
In the system {x+y=4}\{x + y = 4\} with 22 variables, how many degrees of freedom?

Example 25

easy
After fixing the constraint y=2xy = 2x in (x,y)(x, y), how many degrees of freedom remain?

Example 26

easy
In the system {3x−y=0, 6x−2y=0}\{3x - y = 0,\ 6x - 2y = 0\}, how many degrees of freedom?

Example 27

medium
A line segment of fixed length in the plane: how many degrees of freedom (position + orientation)?

Example 28

medium
In a sample of size nn used to estimate the sample variance, how many degrees of freedom?

Example 29

medium
A triangle in 3D space (three vertices in R3\mathbb{R}^3): how many degrees of freedom does the triangle have?

Example 30

medium
A symmetric 3×33 \times 3 matrix has how many independent entries (DOF)?

Example 31

medium
A circle in 3D space (any orientation): how many degrees of freedom does it have?

Example 32

medium
A point in R4\mathbb{R}^4 subject to x1+x2+x3+x4=1x_1 + x_2 + x_3 + x_4 = 1 has how many degrees of freedom?

Example 33

medium
A polynomial of degree ≤3\le 3 in xx has how many degrees of freedom (its coefficients)?

Example 34

hard
In a chi-square goodness-of-fit test with k=5k = 5 categories and no extra estimated parameters, how many degrees of freedom?

Example 35

hard
A point lies on both the plane x+y+z=6x + y + z = 6 and the sphere x2+y2+z2=14x^2 + y^2 + z^2 = 14. Generically, how many degrees of freedom?

Example 36

hard
Five points (xi,yi)(x_i, y_i) on a parabola y=ax2+bx+cy = a x^2 + b x + c — how many degrees of freedom in the curve, and is the system over-, exact-, or under-determined?

Example 37

hard
An ellipse in standard position x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 has how many degrees of freedom (general position in the plane, ignoring axis alignment)?

Example 38

hard
In a 2×22\times 2 contingency table for a chi-square test of independence, how many degrees of freedom?

Example 39

challenge
A general rotation in 3D space (orthogonal matrix with determinant +1+1) has how many degrees of freedom?

Example 40

challenge
A two-sample tt-test using pooled variance with sample sizes n1=10n_1 = 10 and n2=14n_2 = 14 has how many degrees of freedom?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

systems of equationsconstraints