Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

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.

$$\mathrm{2}\:\mathrm{solutions}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com