Question

The value (s) of m does the system of equations 3x + my = m and 2x – 5y = 20 has a solution satisfying the conditions x > 0, y > 0.

Options

• `m ∈ (0, oo)`

• `m ∈ (- oo, - 15/2) ∪ (30, oo)`

• `m ∈ (- 15/2, oo)`

• None of these

Answer 1

`m ∈ (- oo, - 15/2) ∪ (30, oo)`

Explanation:

By using Cramer's rule, the solution of the system is `x = (Δ_x)/Δ, y = (Δ_y)/Δ`, 

Where Δ = `|(3, m),(2, -5)|` = (15 + 2 m)

`Δ_x = |(m, m),(20, -5)|` = – 25 m

`Δ_y = |(3m, m),(2, 20)|` = 60 – 2 m

⇒ `x = (- 25  m)/(- (15 + 2 m)) = (25  m (15 + 2  m))/(15 + 2  m)^2` > 0,

For  m > 0 or m < `- 15 / 2`

Also, y = `(60 - 2  m)/(- (15 + 2  m)) = (2(m - 30)(15 + 2  m))/(15 + 2  m)^2` > 0

For m > 30 or m < `- 15/2`

⇒ x > 0, y > 0 for m > 30 or m = `- 15/2`

For m = `- 15/2`, the system has no solution.