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Alternative-forms-x-y-23-12-9x-16y-29-




Question Number 117555 by dw last updated on 12/Oct/20
Alternative forms   { (((√x)+(√y)=((23)/(12)))),((9x+16y=29)) :}
$${Alternative}\:{forms} \\ $$$$\begin{cases}{\sqrt{{x}}+\sqrt{{y}}=\frac{\mathrm{23}}{\mathrm{12}}}\\{\mathrm{9}{x}+\mathrm{16}{y}=\mathrm{29}}\end{cases} \\ $$$$ \\ $$
Answered by Olaf last updated on 12/Oct/20
(√y) = ((23)/(12))−(√x)  9x+16(((23)/(12))−(√x))^2  = 29  9x+((529)/9)−((184)/3)(√x)+16x = 29  25x−((184)/3)(√x)+((268)/9) = 0  Δ = (((184)/3))^2 −4×25×((268)/9) = 784 = 28^2   (√x) = ((((184)/3)±28)/(50)) = (2/3) or ((134)/(75))  ⇒ x = (4/9) or ((17956)/(5625))  y = ((29−9x)/(16)) = ((23)/(16)) or ((169)/(10000))  S = {((2/3);((23)/(16))),(((134)/(75));((169)/(10000)))}
$$\sqrt{{y}}\:=\:\frac{\mathrm{23}}{\mathrm{12}}−\sqrt{{x}} \\ $$$$\mathrm{9}{x}+\mathrm{16}\left(\frac{\mathrm{23}}{\mathrm{12}}−\sqrt{{x}}\right)^{\mathrm{2}} \:=\:\mathrm{29} \\ $$$$\mathrm{9}{x}+\frac{\mathrm{529}}{\mathrm{9}}−\frac{\mathrm{184}}{\mathrm{3}}\sqrt{{x}}+\mathrm{16}{x}\:=\:\mathrm{29} \\ $$$$\mathrm{25}{x}−\frac{\mathrm{184}}{\mathrm{3}}\sqrt{{x}}+\frac{\mathrm{268}}{\mathrm{9}}\:=\:\mathrm{0} \\ $$$$\Delta\:=\:\left(\frac{\mathrm{184}}{\mathrm{3}}\right)^{\mathrm{2}} −\mathrm{4}×\mathrm{25}×\frac{\mathrm{268}}{\mathrm{9}}\:=\:\mathrm{784}\:=\:\mathrm{28}^{\mathrm{2}} \\ $$$$\sqrt{{x}}\:=\:\frac{\frac{\mathrm{184}}{\mathrm{3}}\pm\mathrm{28}}{\mathrm{50}}\:=\:\frac{\mathrm{2}}{\mathrm{3}}\:\mathrm{or}\:\frac{\mathrm{134}}{\mathrm{75}} \\ $$$$\Rightarrow\:{x}\:=\:\frac{\mathrm{4}}{\mathrm{9}}\:\mathrm{or}\:\frac{\mathrm{17956}}{\mathrm{5625}} \\ $$$${y}\:=\:\frac{\mathrm{29}−\mathrm{9}{x}}{\mathrm{16}}\:=\:\frac{\mathrm{23}}{\mathrm{16}}\:\mathrm{or}\:\frac{\mathrm{169}}{\mathrm{10000}} \\ $$$$\mathcal{S}\:=\:\left\{\left(\frac{\mathrm{2}}{\mathrm{3}};\frac{\mathrm{23}}{\mathrm{16}}\right),\left(\frac{\mathrm{134}}{\mathrm{75}};\frac{\mathrm{169}}{\mathrm{10000}}\right)\right\} \\ $$
Answered by Olaf last updated on 12/Oct/20
Commented by Olaf last updated on 12/Oct/20
2 solutions.
$$\mathrm{2}\:\mathrm{solutions}. \\ $$

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