How do you know when an equation has infinitely many solution and when an equation has no solution?

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The issue is not when an equation has ... but when a system of equations has an infinite number of solutions, no solution,or a single solution.

TAke two lines:
y=m1* x + b1 and
y=m2*x+b2

If m1=m2, the lines are parallel, never intersect, and have no common points. It is said to have"no solution"
IF m1=m2 and b1=b2, the lines are the same, so have an infinite number of "solutions"

If m1 doesn not equal m2, the lines have a single solution.