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Question # Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists {(5x,+,8y,-,6y,=,14),(3x,+,4y,-,2z,=,8),(x,+,2y,-,2z,=,3):}

Forms of linear equations
ANSWERED Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists
$${(5x,+,8y,-,6y,=,14),(3x,+,4y,-,2z,=,8),(x,+,2y,-,2z,=,3):}$$ 2020-11-17

consider the following system of linear equations:
$$5x+8y-6z=14$$
$$3x+4y-2z=8$$
$$x+2y-2z=3$$
convert into augmented matrix
$$[(5,8,-6,|,14),(3,4,-2,|,8),(1,2,-2,|,3)]$$
Transform the above matrix into reduced row echelon form
$$R_2\Rightarrow\frac{5}{3} R_2-R_1$$
$$R_3\Rightarrow5R_3-R_1$$
$$[(5,8,-6,|,14),(0,-\frac{4}{3},-\frac{8}{3},|,-\frac{2}{3}),(0,2,-4,|,1)]$$
$$R_3\Rightarrow(\frac{2}{3})R_3+R_2$$
$$[(5,8,-6,|,14),(0,-\frac{4}{3},-\frac{8}{3},|,-\frac{2}{3}),(0,0,0,|,0)]$$
Hence, solution of a system of linear equations does not exist