Which of the following conditions gives infinitely many solutions?

Correct answer: (a_1/a_2=b_1/b_2=c_1/c_2)

For two linear equations a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0, infinitely many solutions occur when the two equations represent exactly the same line. The condition for this is a₁/a₂=b₁/b₂=c₁/c₂. Every point on that common line then satisfies both equations, so there are infinitely many common points.

Option A describes intersecting lines and therefore gives one solution. Option B gives parallel distinct lines because the coefficient ratios agree but the constant ratio does not, so it gives no solution. Option C has all three ratios equal and is therefore correct. Option D is not the standard algebraic condition and does not establish coincident lines.