Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 153698 by mathocean1 last updated on 09/Sep/21

solve in R   { ((x^3 −y^3 =19)),((xy=6)) :}

$${solve}\:{in}\:\mathbb{R} \\ $$$$\begin{cases}{{x}^{\mathrm{3}} −{y}^{\mathrm{3}} =\mathrm{19}}\\{{xy}=\mathrm{6}}\end{cases} \\ $$

Answered by amin96 last updated on 09/Sep/21

(x−y)(x^2 +xy+y^2 )=19 ⇒ (x−y)((x−y)^2 +3xy)=19  (x−y)^3 +18(x−y)−19=0  ⇒  x−y=1  (x−y)((x+y)^2 −xy)=19   ⇒  (x+y)^2 =25   { ((x+y=±5)),((x−y=1 )) :}   ⇒   { ((x+y=5)),((x−y=1)) :}  ⇒  { ((x=3)),((y=2)) :}   { ((x+y=−5)),((x−y=1)) :}  ⇒  { ((x=−2)),((y=−3)) :}  ⇒ answer  (3; 2) (−2; −3)

$$\left({x}−{y}\right)\left({x}^{\mathrm{2}} +{xy}+{y}^{\mathrm{2}} \right)=\mathrm{19}\:\Rightarrow\:\left({x}−{y}\right)\left(\left({x}−{y}\right)^{\mathrm{2}} +\mathrm{3}{xy}\right)=\mathrm{19} \\ $$$$\left({x}−{y}\right)^{\mathrm{3}} +\mathrm{18}\left({x}−{y}\right)−\mathrm{19}=\mathrm{0}\:\:\Rightarrow\:\:{x}−{y}=\mathrm{1} \\ $$$$\left({x}−{y}\right)\left(\left({x}+{y}\right)^{\mathrm{2}} −{xy}\right)=\mathrm{19}\:\:\:\Rightarrow\:\:\left({x}+{y}\right)^{\mathrm{2}} =\mathrm{25} \\ $$$$\begin{cases}{{x}+{y}=\pm\mathrm{5}}\\{{x}−{y}=\mathrm{1}\:}\end{cases}\:\:\:\Rightarrow\:\:\begin{cases}{{x}+{y}=\mathrm{5}}\\{{x}−{y}=\mathrm{1}}\end{cases}\:\:\Rightarrow\:\begin{cases}{{x}=\mathrm{3}}\\{{y}=\mathrm{2}}\end{cases} \\ $$$$\begin{cases}{{x}+{y}=−\mathrm{5}}\\{{x}−{y}=\mathrm{1}}\end{cases}\:\:\Rightarrow\:\begin{cases}{{x}=−\mathrm{2}}\\{{y}=−\mathrm{3}}\end{cases}\:\:\Rightarrow\:{answer}\:\:\left(\mathrm{3};\:\mathrm{2}\right)\:\left(−\mathrm{2};\:−\mathrm{3}\right) \\ $$

Answered by mr W last updated on 09/Sep/21

x^3 −((6/x))^3 =19  x^6 −19x^3 −216=0  x^3 =−8 or 27  ⇒x=−2 or 3  ⇒y=−3 or 2

$${x}^{\mathrm{3}} −\left(\frac{\mathrm{6}}{{x}}\right)^{\mathrm{3}} =\mathrm{19} \\ $$$${x}^{\mathrm{6}} −\mathrm{19}{x}^{\mathrm{3}} −\mathrm{216}=\mathrm{0} \\ $$$${x}^{\mathrm{3}} =−\mathrm{8}\:{or}\:\mathrm{27} \\ $$$$\Rightarrow{x}=−\mathrm{2}\:{or}\:\mathrm{3} \\ $$$$\Rightarrow{y}=−\mathrm{3}\:{or}\:\mathrm{2} \\ $$

Commented by mathocean1 last updated on 09/Sep/21

thanks sirs

$${thanks}\:{sirs} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com