Question Number 207092 by hardmath last updated on 06/May/24
$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{x}\:−\:\mathrm{6y}}\\{\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{y}\:−\:\mathrm{6x}}\end{cases}\:\:\:\:\:\Rightarrow\:\:\:\:\mathrm{x}\:+\:\mathrm{y}\:=\:? \\ $$
Answered by A5T last updated on 06/May/24
$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} ={x}−{y}−\mathrm{6}{y}+\mathrm{6}{x}=\mathrm{7}{x}−\mathrm{7}{y} \\ $$$$\Rightarrow\left({x}−{y}\right)\left({x}+{y}\right)=\mathrm{7}\left({x}−{y}\right) \\ $$$${x}−{y}\neq\mathrm{0}\:\Rightarrow{x}+{y}=\mathrm{7} \\ $$$${when}\:{x}={y};\:{x}^{\mathrm{2}} ={y}^{\mathrm{2}} =−\mathrm{5}{x}\Rightarrow{x}={y}=\mathrm{0}\:{or}\:{x}={y}=−\mathrm{5} \\ $$$$\Rightarrow{x}+{y}=\mathrm{0}\:{or}\:{x}+{y}=−\mathrm{10} \\ $$$$\Rightarrow{x}+{y}\in\left\{−\mathrm{10},\mathrm{0},\mathrm{7}\right\} \\ $$
Answered by mr W last updated on 06/May/24
$$\left({i}\right)+\left({ii}\right): \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =−\mathrm{5}\left({x}+{y}\right)\:\Rightarrow{x}+{y}\leqslant\mathrm{0} \\ $$$$ \\ $$$$\left({i}\right)−\left({ii}\right): \\ $$$$\left({x}+{y}\right)\left({x}−{y}\right)=\mathrm{7}\left({x}−{y}\right) \\ $$$${x}\neq{y}:\:\Rightarrow{x}+{y}=\mathrm{7}>\mathrm{0}\:\Rightarrow{rejected} \\ $$$${x}={y}:\:{x}^{\mathrm{2}} =−\mathrm{5}{x}\:\Rightarrow{x}=\mathrm{0}\:{or}\:−\mathrm{5}\: \\ $$$$\Rightarrow{x}+{y}=\mathrm{0}\:{or}\:−\mathrm{10} \\ $$
Commented by Frix last updated on 06/May/24
$$\mathrm{But}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{system}\:\mathrm{are} \\ $$$${x}={y}=\mathrm{0}\:\Rightarrow\:{x}+{y}=\mathrm{0} \\ $$$${x}={y}=−\mathrm{5}\:\Rightarrow\:{x}+{y}=−\mathrm{10} \\ $$$${x}=\frac{\mathrm{7}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{119}}}{\mathrm{2}}\mathrm{i}\wedge{y}=\mathrm{conj}\:{x}\:\Rightarrow\:{x}+{y}=\mathrm{7} \\ $$
Commented by mr W last updated on 06/May/24
$$\left.{i}\:{live}\:{in}\:\mathbb{R}\:{world}\::\right) \\ $$
Commented by Frix last updated on 06/May/24
...but the real world is complex... ��
Commented by mr W last updated on 06/May/24
$${indeed}!\:{the}\:{real}\:{world}\:{is}\:{too} \\ $$$${complex}\:{to}\:{be}\:{real}\:{to}\:{me}… \\ $$