Find the solution set of triplets ( x , y , z ) that fulfil this system using Gauss-

juanberrio8a

juanberrio8a

Answered question

2022-06-19

Find the solution set of triplets ( x , y , z ) that fulfil this system using Gauss-Jordan:
{ x + 2 z = 0 3 x 6 z = 0 2 x 4 z = 0
First of all, I don't see any y variable there. I suppose it doesn't matter and I proceed normally:
[ 1 2 0 3 6 0 2 4 0 ]
f 1
[ 1 2 0 3 6 0 2 4 0 ]
3 f 1 + f 2
[ 1 2 0 0 0 0 2 4 0 ]
2 f 1 + f 3
[ 1 2 0 0 0 0 0 0 0 ]
So, this is the staggered reduced form.
This is an homogeneous system (because of the null column), thus one solution is ( 0 , 0 , 0 ).
Other than that, I have to check out the range of the system. The range is 1, which is less than the number of columns... what is the number of columns?

Answer & Explanation

humusen6p

humusen6p

Beginner2022-06-20Added 22 answers

You are almost at the end.The number of parameters is 3 1 = 2, so we have two parameter say y = t , z = s and hence x = 2 s so the set of solutions are of the form
( x , y , z ) = ( 2 s , t , s ) = ( 0 , 0 , 0 ) + s ( 2 , 0 , 1 ) + t ( 0 , 1 , 0 )
which is a plane or 2 dimensional subspace
polivijuye

polivijuye

Beginner2022-06-21Added 16 answers

You have two variables. and three equations. Solve for x and z using any two, get 0 , 0. Confirm that it is consistent with your third.
Now deduce that all triples ( x , y , z ) = ( 0 , a , 0 ) where a R satisfy this, because your equations don't depend on y.

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