Question

Which of the following system of equations has a unique solution?

Options

• x = 0, x = 1

• x + y = 0, 2x + 2y = 0

• x + y = 2, x – y = 3

• x + y = 5, x + y = 10

Answer 1

x + y = 2, x – y = 3

Explanation:

For a unique solution the two linear equations must be independent their coefficient ratios satisfy `(a_1)/(a_2) ≠ (b_1)/(b_2)`, so they represent intersecting lines.

(A) x = 0 and x = 1 contradict each other no solution.

(B) x + y = 0 and 2x + 2y = 0 are dependent second is 2 × first infinitely many solutions.

(C) x + y = 2 and x – y = 3 are independent.

Add them: 2x = 5

⇒ `x = 5/2`

Then `y = 2 - 5/2`

= `(-1)/2` 

Unique solution `x = 5/2, y = (-1)/2`.

(D) x + y = 5 and x + y = 10 are inconsistent no solution.