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Question Number 908 by 112358 last updated on 20/Apr/15

Show that for the system of   equations                          x+y+z=3                   2x+2y+2z=6                   3x+3y+3z=9  the general solution is given by                            x=λ+1                            y=μ+1                            z=−λ−μ+1  where λ,μ∈R .

$${Show}\:{that}\:{for}\:{the}\:{system}\:{of}\: \\ $$$${equations} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}+{y}+{z}=\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}{x}+\mathrm{2}{y}+\mathrm{2}{z}=\mathrm{6} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}{x}+\mathrm{3}{y}+\mathrm{3}{z}=\mathrm{9} \\ $$$${the}\:{general}\:{solution}\:{is}\:{given}\:{by} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}=\lambda+\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}=\mu+\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{z}=−\lambda−\mu+\mathrm{1} \\ $$$${where}\:\lambda,\mu\in\mathbb{R}\:. \\ $$$$ \\ $$$$ \\ $$

Commented by prakash jain last updated on 21/Apr/15

Only one equation is given  x+y+z=3  Two variable can be independently chosen.  Any value can be chosen for x and y.  If x=λ+1, y=μ+1  x+y+z=3  z=3−x−y=1−λ−μ  λ, μ can be real or complex.  For the general solution λ,μ need not be real.

$$\mathrm{Only}\:\mathrm{one}\:\mathrm{equation}\:\mathrm{is}\:\mathrm{given} \\ $$$${x}+{y}+{z}=\mathrm{3} \\ $$$$\mathrm{Two}\:\mathrm{variable}\:\mathrm{can}\:\mathrm{be}\:\mathrm{independently}\:\mathrm{chosen}. \\ $$$$\mathrm{Any}\:\mathrm{value}\:\mathrm{can}\:\mathrm{be}\:\mathrm{chosen}\:\mathrm{for}\:{x}\:{and}\:{y}. \\ $$$$\mathrm{If}\:{x}=\lambda+\mathrm{1},\:{y}=\mu+\mathrm{1} \\ $$$${x}+{y}+{z}=\mathrm{3} \\ $$$${z}=\mathrm{3}−{x}−{y}=\mathrm{1}−\lambda−\mu \\ $$$$\lambda,\:\mu\:\mathrm{can}\:\mathrm{be}\:\mathrm{real}\:\mathrm{or}\:\mathrm{complex}. \\ $$$$\mathrm{For}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\lambda,\mu\:\mathrm{need}\:\mathrm{not}\:\mathrm{be}\:\mathrm{real}. \\ $$

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