Consider the non-homogenous system of linear equations: 1 2 1 2 1 2 18 5 23 6 2 8 9 3 12 x x x x x x Show that this system is consider (i.e. a solution exists) and the solutions is unique. Find the solution.
Added by Michael H.
Step 1
e., a solution exists) and if the solution is unique. The system is given by: \[ \begin{cases} x_1 + 2x_2 = 18 \\ 6x_1 + 2x_2 = 8 \\ 9x_1 + 3x_2 = 12 \end{cases} \] Show more…
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Consider the following two systems of equations: $$ \begin{array}{rlrl}{5 x_{1}+x_{2}-3 x_{3}} & {=0} & {5 x_{1}+x_{2}-3 x_{3}} & {=0} \\ {-9 x_{1}+2 x_{2}+5 x_{3}} & {=1} & {-9 x_{1}+2 x_{2}+5 x_{3}} & {=5} \\ {4 x_{1}+x_{2}-6 x_{3}} & {=9} & {4 x_{1}+x_{2}-6 x_{3}} & {=45}\end{array} $$ It can be shown that the first system has a solution. Use this fact and the theory from this section to explain why the second system must also have a solution. Make no row operations.)
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