Question

If the system of linear equations

2x + 2y + 3z = a

3x – y + 5z = b

x – 3y + 2z = c

Where a, b, c are non-zero real numbers, has more than one solution, then:
1. b – c + a = 0
2. b – c – a = 0
3. a + b + c = 0
4. b + c – a = 0

Answer 1

Correct Answer - Option 2 : b – c – a = 0

If the system of equations has more than one solution, then D = 0 and D₁ = D₂ = D₃ = 0.

From the given set of equation, we form,

D₁ = 0

\(\Rightarrow \left| \begin{matrix} a & 2 & 3 \\ b & -1 & 5 \\ c & -3 & 2 \\ \end{matrix} \right|=0\) 

Now,

⇒ a(-2 + 15) – 2(2b – 5c) + 3(-3b + c) = 0

⇒ 13a – 4b + 10c – 9b + 3c = 0

⇒ 13a – 13b + 13c = 0

⇒ a – b + c = 0

∴ b – a – c = 0