For the pair a₁x + b₁y = c₁ and a₂x + b₂y = c₂, compare the ratios. If a₁/a₂ ≠ b₁/b₂, one solution. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, no solution. If all three ratios equal, infinite solutions.
I explain this to my daughter for her JEE, drawing lines on her notebook with my old gamcha beside us. But you know, in my NGO work, we often face equations with no solution—like when the funder's promise and the ground reality never meet, no matter how you adjust the variables.
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