Problems from Equivalent systems

Write a system of linear equations with 5 unknowns and 3 equations, and using the 5 given rules separately create equivalent systems.

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Development:

First of all we write a system of 5 unknowns and only 3 equations: {2x+3y−z+u−2v=1x−3y+2z−8+2v=−1−2x+y−2z−u+2v=3

or we can also write it in the matricial form: (23−11−2|11−3211|−1−21−2−12|3)

1) It is unimportant: 2x+3y−z+u−2v=1 (2x+3y−z+u−2v)+3x-2=1+3x-2

2)We multiply a line by a number other than zero: (23−11−2|11−3211|−1−21−2−12|3)⇒5⋅row1⇒(1015−55−10|51−3211|−1−21−2−12|3) and we obtain an equivalent system

3) Now we add up row2 to row1 to obtain an equivalent system: (23−11−2|11−3211|−1−21−2−12|3)⇒row1+row2⇒(30−12−1|01−3211|−1−21−2−12|3)

4) Now we add a linear combination of the lines 2 and 3 to row1 to obtain an equivalent system: (23−11−2|11−3211|−1−21−2−12|3)⇒row1−2⋅row2+row3⇒(−210−7−2−2|61−3211|−1−21−2−12|3)

5) Finally a simple change of order also gives us an equivalent system (23−11−2|11−3211|−1−21−2−12|3)⇒col2↔col3⇒(2−131−2|112−311|−1−2−21−12|3)

Solution:

1) Trivial

2) (1015−55−10|51−3211|−1−21−2−12|3)

3) (30−12−1|01−3211|−1−21−2−12|3)

4) (−210−7−2−2|61−3211|−1−21−2−12|3)

5) (2−131−2|112−311|−1−2−21−12|3)

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