The augmented matrix of a set of linear equations in the variables x, y, and z is shown below. It's presented in condensed row-echelon form.[10−1301250000]&nbsp;Find: (a) The leading variables, (b) Is the system in consistent or dependent? (c) The solution of the system.

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The augmented matrix of a set of linear equations in the variables x, y, and z is shown below. It&#039;s presented in condensed row-echelon form.[10−1301250000]&amp;nbsp;Find: (a) The leading variables, (b) Is the system in consistent or dependent? (c) The solution of the system.

komunidadO · Accepted Answer

(a) The diagonal term for variable z is zero, as can be seen from the following matrix in row echelon form. The terms for x and y are equal to 1.[10−1301250000]&amp;nbsp;Hence, the leading variables are x and y.&amp;nbsp;(b) z is a non-leading variable, which explains why. Because of this, the linear equation system that is represented by the provided augmented matrix is a dependent system.(c) After the reduced row echelon form, the corresponding system of equations needs to be written and solved using back substitution:&amp;nbsp;System{x−z=3y+2z=50=0&amp;nbsp;&amp;nbsp;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:&amp;nbsp;x−z=3&amp;nbsp;x−t=3&amp;nbsp;x=t+3&amp;nbsp;From second equation:&amp;nbsp;y+2z=5&amp;nbsp;y+2t=5&amp;nbsp;y=5−2t&amp;nbsp;Result:&amp;nbsp;(a) The leading variables are x and y&amp;nbsp;(b) Dependent system&amp;nbsp;(c)&amp;nbsp;x=t+3,y=5−2t,z=t&amp;nbsp;&amp;nbsp;

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.)

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