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

Answer

`m lt -(15)/(12) "or" m gt 30`

Explanation

Given system of equations are
3x +my =m and 2x-5y =20
Here, `Delta = |{:(3, m),(2, -5):}| =-15-2m`
and `Delta_(x) = |{:(m, m), (20, -5):}| = -25m`
`Delta_(y) = |{:(3, m), (2, 20):}| =60-2m`
If `Delta = 0` then system is inconsistent, i.e it has no solution.
If `Delta ne 0, i.e. m ne (15)/(2)`, the system has a unique solution for any fixed value of m.
We have, `x = (Delta_(x))/(Delta) = (-25m)/(-15-2m) = (25m)/(15+2m)`
and `y = (Delta_(y))/(Delta) = (60-2m)/(-15-2m) =(2m -60)/(15+2m)`
For `x gt 0, (25m)/(15+2m) gt 0`
`rArr m gt 0`
or `m lt - (15)/(2) " "...(i)`
and `y lt 0, (2m-60)/(2m+15) gt 0 rArr m gt 30 or m lt -(15)/(2) " "... (ii)`
From Eqs. (i) and (ii), we get `m lt -(15)/(2) "or" m gt 30`