3x + y = 10; 9x + py = 23

(i) 3x + y = 10; 9x + py = 23

Sol. 3x + y = 10

Comparing with a1x + b1y = c1 we get, a1 = 3, b1 = 1, c1 = 10

9x + py = 23

Comparing with a2x + b2y = c2 we get, a2 = 9, b2 = p, c2 = 23

a₁/a₂ = 3/9 = 1/3  ............ eq. no. (i)

b₁/b₂ = 1/p  ............ eq. no. (ii)

c₁/c₂ = 10/23  ............ eq. no. (iii)

We know that the condition for the equations to have unique solution is_

a₁/a₂ ≠ b₁/b₂

∴ 1/3 ≠ 1/p

∴ p ≠ 3

∴ The simultaneous equations will have unique solution for all values of p except 3.