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Consider a system of two linear equations: a1*x + b1*y = c1 and a2*x + b2*y = c2, where all coefficients a1, b1, c1, a2, b2, c2 are non-zero. If the system has infinitely many solutions, which of the following must be true?

1. a1/a2 = b1/b2 = c1/c2
2. (a1 + a2)/(a1 - a2) = (b1 + b2)/(b1 - b2) = (c1 + c2)/(c1 - c2)
3. The quadratic equations a1*x² + b1*x + c1 = 0 and a2*x² + b2*x + c2 = 0 have no common root
4. The system a1²*x + b1²*y = c1²*c2 and a1*a2*x + b1*b2*y = c1*c2² will also have infinitely many solutions

Correct answer: a1/a2 = b1/b2 = c1/c2

Solution

For infinitely many solutions, both equations must represent the same line, so all corresponding coefficients must be in the same ratio: a1/a2 = b1/b2 = c1/c2. This immediately confirms option A. The other options either follow from or contradict standard properties of such systems.

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