how do you know when an equation has infinitely many solution? How do you know when an equation has no solution
I will assume that your are dealing with linear equations
To have an infinite number of solutions, the ratio of coefficients of one equation must be the same as those of the other.
e.g.
4x + 3y = 5 and
8x + 6y = 10 has an infinite number of solutions
to have no solution the two lines would have to be parallel but distinct, which means the x,y coefficients must be the same, but the ratio of the constants has to be different from that ratio.
e.g.
2x - 3y = 9
8x - 12y = 44 have no solution because the two parallel lines never meet.