For the following system of homogeneous linear equations, find the solution if it is unique, otherwise, describe the infinite solution set in terms of an arbitrary parameter k. 4x-2y+6z=0 5x-y-z=0 2x-y+3z=0

naivlingr 2020-11-06 Answered
For the following system of homogeneous linear equations, find the solution if it is unique, otherwise, describe the infinite solution set in terms of an arbitrary parameter k. 4x-2y+6z=0 5x-y-z=0 2x-y+3z=0
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Expert Answer

Willie
Answered 2020-11-07 Author has 95 answers
Step 1
Given linear equations are
4x−2y+6z=0
5x−y−z=0
2x−y+3z=0
Step 2
To find the solution of given linear equations.
First write down the given linear equations in augmented matrix.
((4,-2,6,0),(5,-1,-1,0),(2,-1,3,0))
Now use Gauss Jordan elimination method to convert above matrix,
Perform the operation R1R14andR2R25R1
The matrix becomes,
(1123200117300000)
Perform operation R1R1+12R2
The matrix becomes,
(104300117300000)
Here in the third row we get 0x+0y+0z=0,
This means for any value of z there will be solution of x and y .
Therefore the system of linear equation has infinitely many solutions.
Step 3
From (104300117300000) we get,
x43z=0
y173z=0
Now let z = k,
x43k=0
x=43k
and
y173k=0
y=173k
Therefore solutions are (x,y,z)=(43k,173k,k)
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