Question

Show that the system of equations –x + 2y + 2 = 0 and `1/2x - 1/2y - 1 = 0` has a unique solution.

Answer 1

Given: Equation (1): –x + 2y + 2 = 0

Equation (2): `1/2x - 1/2y - 1 = 0`

Step-wise calculation:

1. Put each into simpler form:

From (1): –x + 2y = –2 

⇒ x – 2y = 2

From (2): `1/2x - 1/2y = 1` 

⇒ Multiply by 2: x – y = 2

2. Subtract the second equation from the first:

(x – 2y) – (x – y) = 2 – 2

⇒ –y = 0 

⇒ y = 0

3. Substitute y = 0 into x – y = 2:

x – 0 = 2

⇒ x = 2

4. Uniqueness check by coefficient criterion:

Write a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 with a₁ = –1, b₁ = 2, a₂ = `1/2`, b₂ = `-1/2`. 

Since `(a_1)/(a_2) = (-1)/(1/2) = -2` and `(b_1)/(b_2) = 2/(-1/2) = -4`, we have `(a_1)/(a_2) ≠ (b_1)/(b_2)`, so the system has a unique solution.

The unique solution is x = 2, y = 0.