Question Number 13986 by RasheedSindhi last updated on 26/May/17
$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{system} \\ $$$$\mathrm{of}\:\mathrm{equations}. \\ $$$$\:\:\:\:\:\:\:\:\frac{\mathrm{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{x}}}+\frac{\sqrt{\mathrm{y}}}{\mathrm{y}^{\mathrm{2}} }=\frac{\mathrm{1729}}{\mathrm{64}} \\ $$$$\:\:\:\:\:\:\:\:\frac{\mathrm{y}^{\mathrm{2}} }{\:\sqrt{\mathrm{x}}}−\frac{\sqrt{\mathrm{y}}}{\mathrm{x}^{\mathrm{2}} }=\frac{\mathrm{6908}}{\mathrm{81}} \\ $$$$ \\ $$
Commented by ajfour last updated on 27/May/17
$$\frac{\mathrm{1729}}{\mathrm{64}}=\frac{\mathrm{1728}}{\mathrm{64}}+\frac{\mathrm{1}}{\mathrm{64}} \\ $$$$\:\:=\frac{\left(\mathrm{12}\right)^{\mathrm{3}} }{\left(\mathrm{4}\right)^{\mathrm{3}} }+\frac{\left(\mathrm{1}\right)^{\mathrm{3}} }{\left(\mathrm{4}\right)^{\mathrm{3}} }\:=\:\:\:\left(\mathrm{3}\right)^{\mathrm{3}} +\frac{\mathrm{1}}{\left(\mathrm{4}\right)^{\mathrm{3}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:=\left(\mathrm{9}\right)^{\mathrm{3}/\mathrm{2}} +\frac{\mathrm{1}}{\left(\mathrm{16}\right)^{\mathrm{3}/\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\left(\mathrm{9}\right)^{\mathrm{2}} }{\:\sqrt{\mathrm{9}}}+\frac{\sqrt{\mathrm{16}}}{\left(\mathrm{16}\right)^{\mathrm{2}} }\:=\frac{{x}^{\mathrm{2}} }{\:\sqrt{{x}}}+\frac{\sqrt{{y}}}{{y}^{\mathrm{2}} } \\ $$$${by}\:{comparing},\:\:\boldsymbol{{x}}=\mathrm{9}\:\:\:{and}\:\:\boldsymbol{{y}}=\mathrm{16}\: \\ $$$$ \\ $$$$\frac{\mathrm{6908}}{\mathrm{81}}=\frac{\mathrm{6912}}{\mathrm{81}}−\frac{\mathrm{4}}{\mathrm{81}} \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\mathrm{256}}{\mathrm{3}}−\frac{\mathrm{4}}{\mathrm{81}}\: \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\left(\mathrm{16}\right)^{\mathrm{2}} }{\:\sqrt{\mathrm{9}}}−\frac{\sqrt{\mathrm{16}}}{\left(\mathrm{9}\right)^{\mathrm{2}} }=\frac{{y}^{\mathrm{2}} }{\:\sqrt{{x}}}−\frac{\sqrt{{y}}}{{x}^{\mathrm{2}} }\: \\ $$$${again}\:{on}\:{comparison}, \\ $$$$\:\:\:\:\boldsymbol{{x}}=\mathrm{9}\:\:{and}\:\:\boldsymbol{{y}}=\mathrm{16}\:\:. \\ $$$$ \\ $$
Commented by RasheedSindhi last updated on 27/May/17
$$\:\:\:\:\:\:\:\:\:\:=\frac{\left(\mathrm{12}\right)^{\mathrm{3}} }{\left(\mathrm{4}\right)^{\mathrm{3}} }+\frac{\left(\mathrm{1}\right)^{\mathrm{3}} }{\left(\mathrm{4}\right)^{\mathrm{3}} } \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\left(\mathrm{9}\right)^{\mathrm{2}} }{\:\sqrt{\mathrm{9}}}+\frac{\sqrt{\mathrm{16}}}{\left(\mathrm{16}\right)^{\mathrm{2}} }\:\:\mathrm{How}? \\ $$$$\mathrm{pl}\:\mathrm{insert}\:\mathrm{some}\:\mathrm{steps} \\ $$$$ \\ $$
Commented by ajfour last updated on 27/May/17
$${did}\:.\:{Sir}\:! \\ $$
Commented by RasheedSindhi last updated on 27/May/17
$$\theta\alpha{nk}\:{you}! \\ $$