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help-me-x-3y-z-2-3x-4y-2z-3-2x-7y-3z-5-Gauss-Method-




Question Number 161429 by greg_ed last updated on 17/Dec/21
help me !   { ((x+3y+z=2)),((−3x+4y+2z=3)),((−2x+7y+3z=5)) :}  Gauss Method...
$$\mathrm{help}\:\mathrm{me}\:! \\ $$$$\begin{cases}{{x}+\mathrm{3}{y}+{z}=\mathrm{2}}\\{−\mathrm{3}{x}+\mathrm{4}{y}+\mathrm{2}{z}=\mathrm{3}}\\{−\mathrm{2}{x}+\mathrm{7}{y}+\mathrm{3}{z}=\mathrm{5}}\end{cases} \\ $$$$\boldsymbol{\mathrm{G}}\mathrm{auss}\:\mathrm{Method}… \\ $$
Commented by 1549442205PVT last updated on 18/Dec/21
the given system of equations is not  compatible because  △= determinant (((1       3     1)),((−3   4    2)),((−2    7   3)))=12−12−21+8+27−14=0  the system is equivalent to   { ((x+3y+z=2)),((13y+5z=9)) :}⇒ { ((x=((2z−1)/(13)))),((y=((9−5z)/(13)))) :}  or { ((x=((2t−1)/(13)))),((y=((9−5t)/(13)))),((z=t)) :}  ∀t∈R
$${the}\:{given}\:{system}\:{of}\:{equations}\:{is}\:{not} \\ $$$${compatible}\:{because} \\ $$$$\bigtriangleup=\begin{vmatrix}{\mathrm{1}\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\mathrm{1}}\\{−\mathrm{3}\:\:\:\mathrm{4}\:\:\:\:\mathrm{2}}\\{−\mathrm{2}\:\:\:\:\mathrm{7}\:\:\:\mathrm{3}}\end{vmatrix}=\mathrm{12}−\mathrm{12}−\mathrm{21}+\mathrm{8}+\mathrm{27}−\mathrm{14}=\mathrm{0} \\ $$$${the}\:{system}\:{is}\:{equivalent}\:{to} \\ $$$$\begin{cases}{{x}+\mathrm{3}{y}+{z}=\mathrm{2}}\\{\mathrm{13}{y}+\mathrm{5}{z}=\mathrm{9}}\end{cases}\Rightarrow\begin{cases}{{x}=\frac{\mathrm{2}{z}−\mathrm{1}}{\mathrm{13}}}\\{{y}=\frac{\mathrm{9}−\mathrm{5}{z}}{\mathrm{13}}}\end{cases} \\ $$$${or\begin{cases}{{x}=\frac{\mathrm{2}{t}−\mathrm{1}}{\mathrm{13}}}\\{{y}=\frac{\mathrm{9}−\mathrm{5}{t}}{\mathrm{13}}}\\{{z}={t}}\end{cases}}\:\:\forall{t}\in{R} \\ $$

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