Well, my boy, you look at a1/a2, b1/b2 and c1/c2. If a1/a2 equals b1/b2 but not c1/c2, lines are parallel—no solution. If all three ratios equal, lines overlap—infinite solutions. Otherwise, one neat intersection point. I learned this from a scanned 1980s textbook PDF; my old printer spat out the page just before the village transformer failed for the evening, so I had to solve problems by lantern light, making the ratios very clear indeed.
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