Matrices and Determinants 4 Question 30

31. 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 $x>0, y>0$ ?

(1979, 3M )

Show Answer

Answer:

Correct Answer: 31. (1)

Solution:

  1. Given system of equations are

$ 3 x+m y=m \quad \text { and } \quad 2 x-5 y=20 $

Here,

$ \begin{aligned} & \Delta=\left|\begin{array}{cc} 3 & m \\ 2 & -5 \end{array}\right|=-15-2 m \\ & \Delta_{x}=\left|\begin{array}{cc} m & m \\ 20 & -5 \end{array}\right|=-25 m \\ & \Delta_{y}=\left|\begin{array}{cc} 3 & m \\ 2 & 20 \end{array}\right|=60-2 m \end{aligned} $

If $\Delta=0$, then system is inconsistent, i.e. it has no solution.

If $\Delta \neq 0$, i.e. $m \neq \frac{15}{2}$, the system has a unique solution for any fixed value of $m$.

We have, $\quad x=\frac{\Delta_{x}}{\Delta}=\frac{-25 m}{-15-2 m}=\frac{25 m}{15+2 m}$

and

$ y=\frac{\Delta_{y}}{\Delta}=\frac{60-2 m}{-15-2 m}=\frac{2 m-60}{15+2 m} $

For $x>0, \frac{25 m}{15+2 m}>0$

$\Rightarrow m>0$

or $m<-\frac{15}{2}$

and $y>0, \frac{2 m-60}{2 m+15}>0 \Rightarrow m>30$ or $m<-\frac{15}{2}$

From Eqs. (i) and (ii), we get $m<-\frac{15}{2}$ or $m>30$



NCERT Chapter Video Solution

Dual Pane