In the cross-multiplication method, the solution exists (is unique) only when:

For a₁$x$ + b₁$y$ + c₁ = 0 and a₂$x$ + b₂$y$ + c₂ = 0, the solution (when unique) is:

$x$ / (b₁c₂ − b₂c₁) = $y$ / (c₁a₂ − c₂a₁) = 1 / (a₁b₂ − a₂b₁).

From these:
$x$ = (b₁c₂ − b₂c₁) / (a₁b₂ − a₂b₁)
$y$ = (c₁a₂ − c₂a₁) / (a₁b₂ − a₂b₁)

**Caveat**: this works ONLY if (a₁b₂ − a₂b₁) ≠ 0 (i.e., lines aren't parallel/coincident).

Useful when you've memorised the formula and want to skip elimination steps.