For what values of $m$ does the system of equations $3 x+m y=m$ and $2 x-5 y=20$ has a solution sati

Question

For what values of $m$ does the system of equations $3 x+m y=m$ and $2 x-5 y=20$ has a solution satisfying the conditions $x>0, y>0$ ?

Professor PC Thomas Classes · Answer

$3 x+m y=m$ $$\begin{equation*} 2 x-5 y=20 ag{1} \end{equation*}$$ Here $\Delta=\left|\begin{array}{cc}3 & m \ 2 & -5\end{array} ight|=-15-2 m$\ and $\Delta x=\left|\begin{array}{cc}m & m \ 20 & -5\end{array} ight|=-25 m$\ $\Delta y=\left|\begin{array}{ll}3 & m \ 2 & 20\end{array} ight|=60-2 m$\ If $\Delta=0$, the system is inconsistent\ i.e. has no solution. If $\Delta eq 0$, i.e. $m eq-\frac{15}{2}$, the system has a unique solution for any fixed value of $m$.\ we have $x=\frac{\Delta x}{\Delta}=\frac{-25 m}{-15-2 m}=\frac{25 m}{15+2 m}$\ $y=\frac{\Delta y}{\Delta}=\frac{60-2 m}{-15-2 m}=\frac{2 m-60}{2 m+15}$\ We see that for $x>0, m>0$ or $m<-\frac{15}{2}$, for $y>0, m>30$ or $m<-\frac{15}{2}$ combining, $m<-\frac{15}{2}$ or $m>30$ for $x>0, y>0$\