Question Number 15051 by tawa tawa last updated on 07/Jun/17
$$\mathrm{Solve}\:\mathrm{simultaneously} \\ $$$$ \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:=\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:…………\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \:=\:\mathrm{92}\:\:\:\:\:\:\:\:\:……….\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{x}\:−\:\mathrm{y}\:=\:\mathrm{z}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:………..\:\mathrm{equation}\:\left(\mathrm{iii}\right) \\ $$
Answered by Tinkutara last updated on 07/Jun/17
$${y}\:+\:{z}\:=\:{x} \\ $$$$\mathrm{2}{x}\:=\:\mathrm{6}\:\Rightarrow\:{x}\:=\:\mathrm{3} \\ $$$${y}\:+\:{z}\:=\:\mathrm{3} \\ $$$${y}^{\mathrm{3}} \:+\:{z}^{\mathrm{3}} \:=\:\mathrm{92}\:−\:\mathrm{27}\:=\:\mathrm{65} \\ $$$$\left({y}\:+\:{z}\right)\left({y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:−\:{yz}\right)\:=\:\mathrm{65} \\ $$$${y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:−\:{yz}\:=\:\frac{\mathrm{65}}{\mathrm{3}}\:=\:\left({y}\:+\:{z}\right)^{\mathrm{2}} \:−\:\mathrm{3}{yz} \\ $$$$\mathrm{9}\:−\:\mathrm{3}{yz}\:=\:\frac{\mathrm{65}}{\mathrm{3}} \\ $$$$\mathrm{4}{yz}\:=\:\frac{−\mathrm{152}}{\mathrm{9}} \\ $$$$\left({y}\:−\:{z}\right)^{\mathrm{2}} \:=\:\left({y}\:+\:{z}\right)^{\mathrm{2}} \:−\:\mathrm{4}{yz}\:=\:\mathrm{9}\:+\:\frac{\mathrm{152}}{\mathrm{9}} \\ $$$${y}\:−\:{z}\:=\:\pm\:\frac{\sqrt{\mathrm{233}}}{\mathrm{3}} \\ $$$${y}\:+\:{z}\:=\:\mathrm{3} \\ $$$${y}\:=\:\frac{\mathrm{9}\:\pm\:\sqrt{\mathrm{233}}}{\mathrm{6}} \\ $$$${z}\:=\:\frac{\mathrm{9}\:\mp\:\sqrt{\mathrm{233}}}{\mathrm{6}} \\ $$$${x}\:=\:\mathrm{3} \\ $$
Commented by tawa tawa last updated on 07/Jun/17
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$