With fewer equations than variables the best that can be hoped for is to define a relationship between them. For example, one equation with 3 variables can be graphically represented as a plane in 3 dimensions.
If another equation is added it means that two equations with three variables establishes a relationship between one variable and the other two, provided there is consistency and the second equation isn't just a restating of the first equation. In other words it doesn't describe the same plane. So graphically the planes will intersect at a line and the planes are not parallel.
This argument can be extended for more variables and equations. So for there to be a solution, there must be consistency which preserves or augments the relationship between the variables.