Practice Redundancy in Math

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

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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}\{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}\{3x - y = 2,\ kx - y = 2\} to be redundant, find kk.

Example 3

easy
Are x+2y=5x + 2y = 5 and 2x+4y=102x + 4y = 10 the same constraint?

Example 4

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

Example 5

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

Example 6

medium
A survey gives {a+b=100, a+b2=50}\{a + b = 100,\ \tfrac{a+b}{2} = 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}\{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}\{x + y + z = 6,\ 2x + 2y + 2z = 12,\ x + y + z = 7\} redundant, contradictory, or both?

Example 9

easy
A 2×22 \times 2 system with one redundant equation: how many unknowns are still free?

Example 10

medium
In {x+y=5, 2x+2y=9}\{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}\{a=2,\ b=5,\ a+b=7\}?

Example 12

hard
A 4×44 \times 4 system has 22 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}\{x+y+z=6,\ 2x+2y+2z=12\}, how many degrees of freedom (3 unknowns)?

Example 14

hard
If det⁡(1236)=0\det\begin{pmatrix}1&2\\3&6\end{pmatrix} = 0, are the rows redundant?

Example 15

easy
Is the equation 0=00 = 0 redundant in any system?

Example 16

easy
Is 4x+6y=84x + 6y = 8 redundant given 2x+3y=42x + 3y = 4?

Example 17

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

Example 18

easy
Is 5x+5y=255x + 5y = 25 redundant given x+y=5x + y = 5?

Example 19

easy
In {x+y=32x+2y=6\begin{cases} x + y = 3 \\ 2x + 2y = 6 \end{cases}, is the second equation redundant?

Example 20

easy
Is y=2x+1y = 2x + 1 redundant in {y=2x+1y=−x+4\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}?
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