Question

How to use cross multiplication method in unique solution?

Answer 1

Therefore, y = (c₂a₁ - c₁a₂)/(b₁a₂ - b₂a₁) = (c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁) where (a₁b₂ - a₂b₁) ≠ 0

Therefore, y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁), ------------- (iii)

Again, multiplying both sides of (i) and (ii) by b₂ and b₁ respectively, we get;

a₁b₂x + b₁b₂y + b₂c₁ = 0

a₂b₁x + b₁b₂y + b₁c₂ = 0

Subtracting, a₁b₂x - a₂b₁x + b₂c₁ - b₁c₂ = 0

or, x(a₁b₂ - a₂b₁) = (b₁c₂ - b₂c₁)

or, x = (b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁)

Therefore, x/(b₁c₂ - b₂c₁) = 1/(a₁b₂ - a₂b₁) where (a₁b₂ - a₂b₁) ≠ 0 -------------- (iv)

From equations (iii) and (iv), we get:

x/(b₁c₂ - b₂c₁) = y/(c₁a₂) - c₂a₁ = 1/(a₁b₂ - a₂b₁) where (a₁b₂ - a₂b₁) ≠ 0

Hope this helps ☺️☺️