The following matrix is the augmented matrix of a system of linear equations in the variables x, y, and z. (It is given in reduced row-echelon form.)

Wotzdorfg 2020-12-03 Answered
The following matrix is the augmented matrix of a system of linear equations in the variables x, y, and z. (It is given in reduced row-echelon form.)
[101301250000]
Find: (a) The leading variables, (b) Is the system in consistent or dependent? (c) The solution of the system.
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komunidadO
Answered 2020-12-04 Author has 86 answers
(a) It can be seen from the following matrix in row echelon form that the diagonal term corresponding to variable z is zero. Terms corresponding to x and y are equal to 1.
[101301250000]
Hence, the leading variables are x and y.
(b) Because the variable z is a non-leading variable. This makes the system of linear equations represented by the given augmented matrix a dependent system.
(c) After the reduced row echelon form, the corresponding system of equations needs to be written and solved using back substitution:
System{xz=3y+2z=50=0
Since we know that z is a non leading variable we need to use it as a parameter t. So assuming z=t. we have from first equation:
xz=3
xt=3
x=t+3
From second equation: y+2z=5
y+2t=5
y=52t
Result:
(a) The leading variables are x and y
(b) Dependent system
(c) x=t+3,y=52t,z=t
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