The row of zeros at the bottom immediately lets us conclude "infinitely many solutions". This system is consistent and dependent. This is because we have 0x+0y+0z = 0 aka 0 = 0 which is always true for any choice of x, y, and z.
The second row of values lead to the equation 0x+0y+1z = 6, aka z = 6
The first row says: 1x+6y+0z = 1 which is the same as x+6y = 1
Let's say we isolate x
x+6y = 1
x+6y-6y = 1-6y
x = 1-6y
Then we have these three values or expressions
- x = 1-6y
- y = any real number
- z = 6
All of the infinitely many solutions are of the form (x,y,z) = (1-6y, y, 6)
A lot of textbooks will use a parameter such as t, so we could write it as (1-6t, t, 6).
The choice of the letter for the parameter does not matter. I think t is most popular because it represents time. Each (x,y,z) point could represent a particle's location at any given time.
- If t = 0, then we have (1, 0, 6)
- If t = 1, then we have (-5, 1, 6)
- If t = 2, then we have (-11, 2, 6)
- If t = 3, then we have (-17, 3, 6)
and so on.
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Conclusion:
There are infinitely many solutions of the form (x,y,z) = (1-6t, t, 6) where t is any real number.
This system is consistent and dependent.