Question

The system of linear equations

3x – 2y – kz = 10

2x – 4y – 2z = 6

x + 2y – z = 5m

is inconsistent if ______.

Options

• `k = 3, m = 4/5`

• `k ≠ 3, m ∈ R`

• `k ≠ 3, m ≠ 4/5`

• `k = 3, m ≠ 4/5`

Answer 1

The system of linear equations

3x – 2y – kz = 10

2x – 4y – 2z = 6

x + 2y – z = 5m

is inconsistent if `underlinebb(k = 3, m ≠ 4/5)`.

Explanation:

Given: System of linear equations

3x – 2y – kz = 10

2x – 4y – 2z = 6

x + 2y – z = 5m

is inconsistent.

Form determinant D with coefficients of x, y, z.

D_x is formed by replacing coefficients of x with constant terms.

D_y is formed by replacing coefficients of y with constant terms.

D_z is formed by replacing coefficients of z with constant terms.

For, the equation to be inconsistent,

D = `|(3, -2, -k),(2, -4, -2),(1, 2, -1)|` = 0

⇒ 3(4 + 4) + 2(–2 + 2) – k(4 + 4) = 0

⇒ k = 3

Also, at least one of D_x, D_y, D_z must be non-zero.

D_x = `|(10, -2, -3),(6, -4, -2),(5m, 2, -1)|` ≠ 0

⇒ 10(4 + 4) + 2(–6 + 10m) – 3(12 + 20m) ≠ 0

⇒ 80 – 12 + 20m – 36 – 60 m

⇒ 40m ≠ 32

⇒ `m ≠ 4/5`

D_y = `|(3, 10, -3),(2, 6, -2),(1, 5m, -1)|`

= `-|(3, 10, 3),(2, 6, 2),(1, 5m, 1)|` = 0  ...(∵ Two columns of D_y are same)

D_z = `|(3, -2, 10),(2, -4, 6),(1, 2, 5m)|` ≠ 0

⇒ 3(–20m – 12) + 2(10m – 6) + 10(4 + 4) – 40m + 32 ≠ 0

⇒ `m ≠ 4/5`