Question Number 197112 by sonukgindia last updated on 08/Sep/23
Answered by AST last updated on 08/Sep/23
$$\mathrm{2}{x}=\mathrm{1135}−{y}\Rightarrow\mathrm{2}{y}+\mathrm{4}{z}=\mathrm{890}\Rightarrow{y}+\mathrm{2}{z}=\mathrm{445} \\ $$$${x}=\mathrm{345}+{z},{y}=\mathrm{445}−\mathrm{2}{z},{z}={z} \\ $$$$\Rightarrow{No}\:{unique}\:{solutions} \\ $$
Commented by TheHoneyCat last updated on 08/Sep/23
$$\mathrm{In}\:\mathrm{here}\:\mathrm{are}\:\mathrm{all}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{values}: \\ $$$${xyz}=\frac{{x}\left(\mathrm{455}−{y}\right)\left(\mathrm{1135}−{y}\right)}{\mathrm{4}} \\ $$$$\mathrm{and}\:\mathrm{since}\:\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{degree}\:\mathrm{polynomial} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{are}\:\mathbb{R}. \\ $$
Commented by AST last updated on 08/Sep/23
$${Yea},\:{no}\:{single}\:{solution}. \\ $$
Answered by Frix last updated on 09/Sep/23
$$\mathrm{2}{x}=\mathrm{2025}−\mathrm{3}{y}−\mathrm{4}{z} \\ $$$$\mathrm{2}{x}=\mathrm{1135}−{y} \\ $$$$ \\ $$$$\mathrm{2025}−\mathrm{3}{y}−\mathrm{4}{z}=\mathrm{1135}−{y} \\ $$$${y}=\mathrm{445}−\mathrm{2}{z}\:\bigstar \\ $$$$ \\ $$$$\mathrm{2}{x}=\mathrm{1135}−{y}=\mathrm{690}+\mathrm{2}{z} \\ $$$${x}=\mathrm{345}+{z}\:\bigstar \\ $$$$ \\ $$$${z}={c}\in\mathbb{C}\:\bigstar \\ $$$$ \\ $$$${xyz}=\left(\mathrm{345}+{c}\right)\left(\mathrm{445}−\mathrm{2}{c}\right){c} \\ $$