Question Number 54659 by Gulay last updated on 08/Feb/19
Commented by Gulay last updated on 08/Feb/19
$$\mathrm{sir}\:\mathrm{could}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}? \\ $$$$ \\ $$
Commented by maxmathsup by imad last updated on 08/Feb/19
$$\left({s}\right)\:\Leftrightarrow\:\:\begin{cases}{−\mathrm{2}{x}+{y}\:=\mathrm{7}}\\{−{kx}\:+{y}\:=\mathrm{9}}\end{cases} \\ $$$$\Delta_{{s}} =\begin{vmatrix}{−\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{−{k}\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{vmatrix}={k}−\mathrm{2}\:\:\:{so}\:{if}\:{k}\neq\mathrm{2}\:\:\:{the}\:{sy}\:{stem}\:{have}\:{one}\:{solution} \\ $$$$\left({x},{y}\right)\:{with}\:{x}\:=\frac{\Delta_{{x}} }{\Delta}\:=\frac{\begin{vmatrix}{\mathrm{7}\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{9}\:\:\:\:\:\:\:\mathrm{1}}\end{vmatrix}}{{k}−\mathrm{2}}\:=\frac{−\mathrm{2}}{{k}−\mathrm{2}}\:\:{and}\:{y}\:=\frac{\begin{vmatrix}{−\mathrm{2}\:\:\:\:\:\:\mathrm{7}}\\{−{k}\:\:\:\:\:\:\:\mathrm{9}}\end{vmatrix}}{{k}−\mathrm{2}}\:=\frac{\mathrm{7}{k}−\mathrm{18}}{{k}−\mathrm{2}} \\ $$$${if}\:\:{k}\:=\mathrm{2}\:\:\:\:{we}\:{get}\:\:\:\left({s}\right)\:\:\:\begin{cases}{−\mathrm{2}{x}+{y}\:=\mathrm{7}}\\{−\mathrm{2}{x}\:+{y}\:=\mathrm{9}\:\:\:\:\:\:\:\:{and}\:\:{the}\:{system}\:{haven}\:{t}\:{any}\:{solution}\:.}\end{cases} \\ $$
Answered by peter frank last updated on 08/Feb/19
$${y}−{y}−\mathrm{2}{x}−\left(−{kx}\right)=\mathrm{7}−\mathrm{9}=−\mathrm{2} \\ $$$${x}\left({k}−\mathrm{2}\right)=−\mathrm{2} \\ $$$${x}=\frac{−\mathrm{2}}{{k}−\mathrm{2}} \\ $$$${y}=\frac{\mathrm{6}{k}−\mathrm{14}}{{k}−\mathrm{2}} \\ $$