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Question Number 207394 by hardmath last updated on 13/May/24

$$\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}$$$${\mathrm{a}}^3-{\mathrm{b}}^3=625$$$${\mathrm{c}}^3-{\mathrm{d}}^3=1$$$$\mathrm{Find}:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}=?$$

Answered by mr W last updated on 13/May/24

$$\frac{a}{b}=\frac{c}{d}=k,say$$$$a=kb,c=kdwithk\ne 1$$$$b^3(k^3-1)=625\Rightarrow b=\sqrt[3]{\frac{625}{k^3-1}}\Rightarrow a=k\sqrt[3]{\frac{625}{k^3-1}}$$$$d^3(k^3-1)=1\Rightarrow d=\frac{1}{\sqrt[3]{k^3-1}}\Rightarrow c=\frac{k}{\sqrt[3]{k^3-1}}$$$$3equationsfor4variables,so$$$$thereisnouniquesolution.$$

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