Question Number 7713 by Tawakalitu. last updated on 11/Sep/16 | ||
$$\mathrm{2}{x}\:+\:{y}−\:{z}\:=\:\mathrm{8}\:\:\:\:..........\:{equation}\:\left({i}\right) \\ $$$${x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{z}^{\mathrm{2}} \:=\:\mathrm{14}\:\:\:\:..........\:{equation}\:\left({ii}\right) \\ $$$$\mathrm{3}{x}^{\mathrm{3}} \:+\:\mathrm{4}{y}^{\mathrm{3}} \:+\:{z}^{\mathrm{3}} \:=\:\mathrm{195}\:\:\:\:.........\:{equation}\:\left({iii}\right) \\ $$$$ \\ $$$${Solve}\:{simultaneously}. \\ $$ | ||
Commented by Rasheed Soomro last updated on 11/Sep/16 | ||
$${Not}\:{related}\:{to}\:{above} \\ $$$${I}\:{have}\:{corrected}\:{my}\:{answer}\:{to}\:{your}\:{question}\left({Q}#\mathrm{7656}\right) \\ $$$${Pl}\:{see}\:{if}\:{you}\:{have}\:{not}\:{seen}. \\ $$ | ||
Commented by Tawakalitu. last updated on 11/Sep/16 | ||
$${thanks}\:{have}\:{seen}\:{it},\:{i}\:{really}\:{appreciate}..\:{what}\:{of}\:{this}. \\ $$$${simultaneous} \\ $$ | ||
Commented by Rasheed Soomro last updated on 12/Sep/16 | ||
$$\mathrm{2}{x}\:+\:{y}−\:{z}\:=\:\mathrm{8}....................\left({i}\right) \\ $$$${x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{z}^{\mathrm{2}} \:=\:\mathrm{14}..............\left({ii}\right) \\ $$$$\mathrm{3}{x}^{\mathrm{3}} \:+\:\mathrm{4}{y}^{\mathrm{3}} \:+\:{z}^{\mathrm{3}} \:=\:\mathrm{195}............\left({iii}\right) \\ $$$$\left({i}\right)\Rightarrow{z}=\mathrm{2}{x}+{y}−\mathrm{8}\Rightarrow{z}^{\mathrm{6}} =\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{6}} ..........................\left({iv}\right) \\ $$$$\left({ii}\right)\Rightarrow{z}^{\mathrm{2}} =\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\Rightarrow{z}^{\mathrm{6}} =\left(\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right)^{\mathrm{3}} ...\left({v}\right) \\ $$$$\left({iii}\right)\Rightarrow{z}^{\mathrm{3}} =\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} \right)\Rightarrow{z}^{\mathrm{6}} =\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} \right)^{\mathrm{2}} ........\left({vi}\right) \\ $$$$ \\ $$$$\left({iv}\right),\left({v}\right),\left({vi}\right)\Rightarrow \\ $$$$\:\:\:\:\:\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{6}} =\left(\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right)^{\mathrm{3}} =\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} \right)^{\mathrm{2}} \\ $$$$ \\ $$$$\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{6}} =\left(\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right)^{\mathrm{3}} ....................\left({vii}\right) \\ $$$$\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{6}} =\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} \right)^{\mathrm{2}} .........................\left({viii}\right) \\ $$$$\left({vii}\right)\Rightarrow\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{6}} −\left(\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right)^{\mathrm{3}} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\Rightarrow\left[\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} \right]^{\mathrm{3}} −\left(\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right)^{\mathrm{3}} =\mathrm{0} \\ $$$${Let}\:\:\:\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} ={A}\:,\:\:\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]={B} \\ $$$$\:\:\:\:\:\:\:\:\:\:\Rightarrow{A}^{\mathrm{3}} −{B}^{\mathrm{3}} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\Rightarrow\left({A}−{B}\right)\left({A}^{\mathrm{2}} +{AB}+{B}^{\mathrm{2}} \right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\Rightarrow{A}−{B}=\mathrm{0}\:\:\vee\:\:{A}^{\mathrm{2}} +{AB}+{B}^{\mathrm{2}} =\mathrm{0} \\ $$$${A}−{B}=\mathrm{0}\Rightarrow\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} −\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} −\mathrm{14}+{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{2}\left(\mathrm{4}{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{64}+\mathrm{4}{xy}−\mathrm{16}{y}−\mathrm{32}{x}\right)−\mathrm{14}+{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{8}{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} +\mathrm{128}+\mathrm{8}{xy}−\mathrm{32}{y}−\mathrm{64}{x}−\mathrm{14}+{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{9}{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{114}+\mathrm{8}{xy}−\mathrm{32}{y}−\mathrm{64}{x}=\mathrm{0} \\ $$$${A}^{\mathrm{2}} +{AB}+{B}^{\mathrm{2}} =\mathrm{0}\Rightarrow \\ $$$$\:\left\{\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} \right\}^{\mathrm{2}} +\left\{\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} \right\}\left\{\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right\}+\left\{\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right\}^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{4}} +\left\{\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} \right\}\left\{\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right\}+\left(\mathrm{1}/\mathrm{4}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]^{\mathrm{2}} =\mathrm{0} \\ $$$$ \\ $$$$\left({viii}\right)\Rightarrow\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{6}} −\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} \right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\left\{\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{3}} \right\}^{\mathrm{2}} −\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} \right)^{\mathrm{2}} =\mathrm{0} \\ $$$${Let}\:\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{3}} ={a}\:,\:\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} ={b} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow{a}^{\mathrm{2}} −{b}^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\left({a}−{b}\right)\left({a}+{b}\right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:{a}−{b}=\mathrm{0}\:\:\vee\:{a}+{b}=\mathrm{0} \\ $$$${a}−{b}=\mathrm{0}\:\Rightarrow\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{3}} −\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} \right)=\mathrm{0} \\ $$$$ \\ $$$$ \\ $$$${a}+{b}=\mathrm{0}\:\Rightarrow\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{3}} +\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\mathrm{4}{y}^{\mathrm{3}} \right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$$${Continue} \\ $$ | ||
Commented by Tawakalitu. last updated on 12/Sep/16 | ||
$${Thannks}\:{so}\:{much}\:{sir}.\:{am}\:{with}\:{you}\:{sir}. \\ $$ | ||
Commented by Rasheed Soomro last updated on 14/Sep/16 | ||
$$\mathrm{2}{x}\:+\:{y}−\:{z}\:=\:\mathrm{8}....................\left({i}\right) \\ $$$${x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{z}^{\mathrm{2}} \:=\:\mathrm{14}..............\left({ii}\right) \\ $$$$\mathrm{3}{x}^{\mathrm{3}} \:+\:\mathrm{4}{y}^{\mathrm{3}} \:+\:{z}^{\mathrm{3}} \:=\:\mathrm{195}...........\left({iii}\right) \\ $$$$ \\ $$$$\left({i}\right)\Rightarrow\:\:{z}=\mathrm{2}{x}+{y}−\mathrm{8}...................\left({iv}\right) \\ $$$$\left({ii}\right)\Rightarrow{z}^{\mathrm{2}} =\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right].......\left({v}\right) \\ $$$$\left({iii}\right)\Rightarrow{z}^{\mathrm{3}} =\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} ...........\left({vi}\right) \\ $$$$ \\ $$$$\left({ii}\right)\boldsymbol{\div}\left({i}\right)\Rightarrow\:\:{z}=\frac{\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]}{\mathrm{2}{x}+{y}−\mathrm{8}}........\left({vii}\right) \\ $$$$\left({iii}\right)\boldsymbol{\div}\left({ii}\right)\Rightarrow{z}=\frac{\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} }{\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]}........\left({viii}\right) \\ $$$$\left({vii}\right),\left({viii}\right)\Rightarrow\frac{\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]}{\mathrm{2}{x}+{y}−\mathrm{8}}=\frac{\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} }{\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]} \\ $$$$\left\{\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]\right\}^{\mathrm{2}} =\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} \right) \\ $$$$\:\:\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]^{\mathrm{2}} =\mathrm{4}\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} \right)............{A} \\ $$$$ \\ $$$$\left({vi}\right)\boldsymbol{\div}\left({iv}\right)\Rightarrow{z}^{\mathrm{2}} =\frac{\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} }{\mathrm{2}{x}+{y}−\mathrm{8}}..............\left({ix}\right) \\ $$$$\left({v}\right)\Rightarrow\:\:\:\:\:\:\:\:\:\:\:\:\:\:{z}^{\mathrm{2}} =\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right]..........\left({x}\right) \\ $$$$\left({ix}\right),\left({x}\right)\Rightarrow\:\frac{\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} }{\mathrm{2}{x}+{y}−\mathrm{8}}=\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\left(\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} \right)=\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)\left(\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right).....{B} \\ $$$$\left({i}\right)^{\mathrm{2}} \Rightarrow{z}^{\mathrm{2}} =\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} ........................\left({xi}\right) \\ $$$$\left({ii}\right)\Rightarrow{z}^{\mathrm{2}} =\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right].................\left({xii}\right) \\ $$$$\left({xi}\right),\left({xii}\right)\Rightarrow\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} =\left(\mathrm{1}/\mathrm{2}\right)\left[\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{2}\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{2}} =\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{2}\left(\mathrm{4}{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{64}+\mathrm{4}{xy}−\mathrm{16}{y}−\mathrm{32}{x}\right)=\mathrm{14}−{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{8}{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} +\mathrm{128}+\mathrm{8}{xy}−\mathrm{32}{y}−\mathrm{64}{x}−\mathrm{14}+{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{9}{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{8}{xy}−\mathrm{32}{y}−\mathrm{64}{x}+\mathrm{114}...........................{C} \\ $$$$ \\ $$$$\left({iv}\right)^{\mathrm{3}} \Rightarrow{z}^{\mathrm{3}} =\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{3}} ....................\left({xiii}\right) \\ $$$$\left({vi}\right)\Rightarrow\:\:{z}^{\mathrm{3}} =\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} ................\left({xiv}\right) \\ $$$$\left({xiii}\right),\left({xiv}\right)\Rightarrow\left(\mathrm{2}{x}+{y}−\mathrm{8}\right)^{\mathrm{3}} =\mathrm{195}−\mathrm{3}{x}^{\mathrm{3}} −\:\mathrm{4}{y}^{\mathrm{3}} .....................{D} \\ $$$$ \\ $$$$ \\ $$$$\:{Continue}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$ | ||
Commented by Tawakalitu. last updated on 13/Sep/16 | ||
$${Am}\:{with}\:{you}\:{sir}.\:{thanks}\:{for}\:{your}\:{help}. \\ $$ | ||