Practice Redundancy 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

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.

Showing a random 20 of 50 problems.

Example 1

hard
Decide: is {x+y=4, x−y=2, 2x=6} consistent, and if so is the third equation redundant?

Example 2

medium
For {3x−y=2, kx−y=2} to be redundant, find k.

Example 3

easy
Are x+2y=5 and 2x+4y=10 the same constraint?

Example 4

easy
Is {x=1, y=2, 2x+y=4} redundant in any equation?

Example 5

medium
In {x=y, y=z, x=z}, how many are redundant?

Example 6

medium
A survey gives {a+b=100, a+b2=50}. Is the second new information?

Example 7

medium
How many redundant equations are in {x+y=2, 2x+2y=4, 3x+3y=6, 4x+4y=8}?

Example 8

medium
Is {x+y+z=6, 2x+2y+2z=12, x+y+z=7} redundant, contradictory, or both?

Example 9

easy
A 2×2 system with one redundant equation: how many unknowns are still free?

Example 10

medium
In {x+y=5, 2x+2y=9}, is the second redundant or contradictory?

Example 11

medium
Is the third equation redundant: {a=2, b=5, a+b=7}?

Example 12

hard
A 4×4 system has 2 redundant equations and is consistent. How many free parameters does its solution set have?

Example 13

medium
Given {x+y+z=6, 2x+2y+2z=12}, how many degrees of freedom (3 unknowns)?

Example 14

hard
If det⁡(1236)=0, are the rows redundant?

Example 15

easy
Is the equation 0=0 redundant in any system?

Example 16

easy
Is 4x+6y=8 redundant given 2x+3y=4?

Example 17

hard
A constraint set has {x≥1, x≥0, x≤5}. Which is redundant?

Example 18

easy
Is 5x+5y=25 redundant given x+y=5?

Example 19

easy
In {x+y=32x+2y=6, is the second equation redundant?

Example 20

easy
Is y=2x+1 redundant in {y=2x+1y=−x+4?
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