When equations have infinitely many solutions, How would you know this?

Question

also, when working on equations that has no solutions, How would I know this? If posible can you give me examples of each. Thank you

Answer 1

Here is an example of two equations with infinite number of solutions

x+y=4
3x+3y=12

Why infinite? They represent the same line, they share the same set of solutions.

Here is one that has no solutions.

y=3x+5
y=3x+3

Here, the lines are parallel, they have the same slope, but are offset (different intercepts). Therefore they share no common point (solution).