Question

If the system of equations 3x + y = 1 and (2k – 1)x + (k – 1)y = 2k + 1 is inconsistent, then k = ______.

Options

• –1

• 0

• 1

• 2

Answer 1

If the system of equations 3x + y = 1 and (2k – 1)x + (k – 1)y = 2k + 1 is inconsistent, then k = 2.

Explanation:

Given equations

3x + y = 1 ......(1)

(2k – 1)x + (k – 1)y = 2k + 1 ......(2)

3x + y = 1

Comparing with a₁x + b₁y + c₁ = 0

∴ a₁ = 3, b₁ = 1, c₁ = 1

(2k – 1)x + (k – 1)y = 2k + 1

Comparing with a₂x + b₂y + c₂ = 0

∴ a₂ = (2k – 1), b₂ = (k – 1), c₂ = (2k + 1)

Now, 

`a_1/a_2 = 3/((2k - 1))`

`b_1/b_2 = 1/((k - 1))`

`c_1/c_2 = 1/((2k + 1))`

Since equations are inconsistent.

It means that the lines are parallel.

`a_1/a_2 = b_1/b_2 ≠ c_1/c_2`

`3/((2k - 1)) = 1/((k - 1)) ≠ 1/((2k + 1))`

Thus,

`3/((2k - 1)) = 1/((k - 1))`

3(k – 1) = (2k – 1)

3k – 3 = 2k – 1

3k – 2k = –1 + 3

k = 2