Practice Degrees of Freedom in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

hard
A homogeneous linear system in 44 unknowns with rank 44: how many solutions?

Example 4

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 5

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

Example 6

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

Example 7

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

Example 8

medium
A 2-by-3 matrix: how many degrees of freedom in choosing its entries freely?

Example 9

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

Example 10

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?

Example 11

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?

Example 12

easy
A line in 3D space: how many degrees of freedom does a point ON it have?

Example 13

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

Example 14

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 15

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

Example 16

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

Example 17

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

Example 18

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

Example 19

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

Example 20

hard
In a chi-square goodness-of-fit test with k=5k = 5 categories and no extra estimated parameters, how many degrees of freedom?
← Back to Degrees of Freedom Worked Examples Formula Explained