One, none or infinitely many

Not every pair has exactly one answer.

• x – y = 10 and 10x – 10y = 100: multiply the first by 10 and you get the second. Same line, so infinitely many solutions.
• x – y = 10 and 10x – 10y = 101: same left side, different right side. No solution.

The rules, by comparing coefficients:

• a1/a2 ≠ b1/b2 → one solution;
• a1/a2 = b1/b2 = c1/c2 → infinitely many;
• a1/a2 = b1/b2 ≠ c1/c2 → none.

Compare a and b first. Only if a1/a2 = b1/b2 do you need to look at c.

The converses are true too, because exactly one of the three cases always happens.

If a coefficient is 0, fractions like 0/0 cause trouble. Use 'one equation is k times the other' instead.

History. Babylonian clay tablets (c. 2000 BCE) solve such pairs. China's Nine Chapters solved systems with counting rods, using what is really elimination, 18 centuries before Gauss. Āryabhaṭa (499 CE) and Brahmagupta (628 CE) solved them too.