Redundancy Examples in Math

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

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

Concept Recap

An equation in a system that is a linear combination of the others and therefore adds no new constraints or information.

If equation 2 is just equation 1 doubled, it's redundant — the same constraint stated twice.

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: A redundant equation is a combination of the others, so it adds no new information.

Common stuck point: The procedure for redundancy is the easy part; the trap is counting a redundant equation as a real constraint. Asking "Is this equation just a combination of the others, telling me nothing new?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this equation just a combination of the others, telling me nothing new?

Worked Examples

Example 1

easy

\begin{cases} x + y = 3 \\ 2x + 2y = 6 \end{cases}

, is the second equation redundant?

Answer

Yes, equation 2 is redundant.

First step

Step 1: Divide equation 2 by 2:

x + y = 3

Full solution

2
Step 2: This is identical to equation 1.
3
Step 3: Yes, equation 2 adds no new information — it is redundant.

A redundant equation is a scalar multiple of another equation (or a linear combination of other equations). It doesn't reduce degrees of freedom or constrain the solution further.

Example 2

medium

\begin{cases} x + y = 2 \\ 2x - y = 1 \\ 3x = 3 \end{cases}

, is equation 3 redundant?

Example 3

medium

Determine whether

\{2x + 3y = 6,\ 4x + 6y = 11\}

is redundant, contradictory, or independent.

Example 4

medium

A LP has constraints

\{x \ge 0,\ y \ge 0,\ x \ge -3\}

. Which constraint is redundant?

Example 5

hard

Show that Eq3 in

\{x+y=2,\ y+z=3,\ x+2y+z=5\}

is redundant.

Example 6

hard

Decide: is

\{x + y = 4,\ x - y = 2,\ 2x = 6\}

consistent, and if so is the third equation redundant?

Example 7

challenge

\{x+y+z=3,\ x+y=2,\ z=1,\ 2x+2y+2z=6\}

, identify all redundant equations.

Practice Problems

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

Example 1

easy

y = 2x + 1

redundant in

\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}

Example 2

medium

If a

3 \times 3

system has rank 2, how many redundant equations are there?

Example 3

easy

Are

x + 2y = 5

and

2x + 4y = 10

the same constraint?

Example 4

easy

Is the equation

0 = 0

redundant in any system?

Example 5

easy

\{x+y=3,\ 3x+3y=9\}

, how many INDEPENDENT equations are there?

Example 6

easy

Does a redundant equation change the solution set?

Example 7

easy

Two equations describe the same line. Redundant or contradictory?

Example 8

easy

4x + 6y = 8

redundant given

2x + 3y = 4

Example 9

easy

How many UNIQUE constraints in

\{a+b=2,\ 2a+2b=4,\ 3a+3b=6\}

Example 10

easy

If a

2\times2

system has one redundant equation, how many solutions (if consistent)?

Example 11

medium

Is the third equation redundant:

\{x+y=4,\ x-y=2,\ 2x=6\}

Example 12

medium

Given

\{x+y+z=6,\ 2x+2y+2z=12\}

, how many degrees of freedom (3 unknowns)?

Example 13

medium

\{x + y = 5,\ 2x + 2y = 9\}

, is the second redundant or contradictory?

Example 14

medium

A survey gives

\{a + b = 100,\ \tfrac{a+b}{2} = 50\}

. Is the second new information?

Example 15

medium

Which equation can be removed without changing solutions:

\{x=1,\ y=2,\ x+y=3\}

Example 16

medium

\{2x - y = 0,\ -6x + 3y = 0\}

contain a redundant equation?

Example 17

medium

A system of

3

equations in

3

unknowns has

1

redundant equation and is consistent. Unique solution?

Example 18

challenge

Show that in

\{x+y+z=3,\ x+2y+3z=6,\ 2x+3y+4z=9\}

the third equation is redundant.

Example 19

challenge

Find

c

so that

\{x + 2y = 3,\ 2x + 4y = c\}

is redundant (not contradictory).

Example 20

challenge

A linear program lists

\{x+y\le 10,\ 2x+2y\le 20\}

as two constraints. Is one redundant?

Example 21

medium

Is the third equation redundant:

\{a=2,\ b=5,\ a+b=7\}

Example 22

medium

\{x - y = 1,\ 3x - 3y = 3,\ x + y = 5\}

, which equation is redundant?

Example 23

easy

5x + 5y = 25

redundant given

x + y = 5

Example 24

easy

Are

\{y = x + 1,\ y - x = 1\}

redundant?

Example 25

easy

\{x = 1,\ y = 2,\ 2x + y = 4\}

redundant in any equation?

Example 26

easy

True/false: removing a redundant equation may change the solution set.

Example 27

medium

For

\{3x - y = 2,\ kx - y = 2\}

to be redundant, find

k

Example 28

medium

How many redundant equations are in

\{x+y=2,\ 2x+2y=4,\ 3x+3y=6,\ 4x+4y=8\}

Example 29

medium

\{x + y + z = 6,\ 2x + 2y + 2z = 12,\ x + y + z = 7\}

redundant, contradictory, or both?

Example 30

medium

Is the third equation redundant in

\{x+2y=5,\ 2x+y=4,\ 3x+3y=9\}

Example 31

hard

For what value of

k

\{x + 2y - z = 3,\ 2x + 4y - 2z = k\}

redundant?

Example 32

hard

4 \times 4

system has

2

redundant equations and is consistent. How many free parameters does its solution set have?

Example 33

hard

A coefficient matrix has

\det = 0

for a square system with consistent RHS. Are some equations redundant?

Example 34

hard

True/false: redundancy in equations means the matrix is singular.

Example 35

hard

A constraint set has

\{x \ge 1,\ x \ge 0,\ x \le 5\}

. Which is redundant?

Example 36

challenge

For

\{ax + by = c,\ 2ax + 2by = 2c + 1\}

to be contradictory, what condition on

a, b, c

Example 37

challenge

A real-life setup: a recipe lists both "makes 4 servings" and "each serving = 1/4 of the batch." Which is redundant?

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Related Concepts

Systems of Equations Consistency

Background Knowledge

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

systems of equations