LINEAR SYSTEM

$\begin{array}{rlrlrlrlrlrlrlrlrlrlr}{x}_1& & +& & 6x_2& & & & & & +& & 3x_4& & & & & & =& & 0\\ & & & & & & & & x_3& & -& & 8x_4& & & & & & =& & 5\\ & & & & & & & & & & & & & & & & x_5& & =& & -9\end{array}$

FORMS AND HOMOGENEITY

$\mathbf{Ax}\mathbf{=}\mathbf{b}$

ROW REDUCE

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CONSISTENCY AND SOLUTION SET

The system is consistent because its row echelon augmented matrix contains no pivot position in its final column.

The system has infinitely many solutions because its row echelon coefficient matrix contains at least one pivot-free column.

PARAMETRIC REPRESENATION OF SOLUTION SET

$\begin{array}{rl}{x}_1+6x_2+3x_4=0⟺x_1& =-6x_2-3x_4\\ x_2& \text{free}\\ x_3-8x_4=5⟺x_3& =8x_4+5\\ x_4& \text{free}\\ x_5& =-9\end{array}$

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LINEAR INDEPENDENCE

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GEOMETRIC INTERPRETATION OF SOLUTION SET

The solution set is a plane through the point $(0,0,5,0,-9)$ which is parallel to

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