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1 See Before: homogeneous equations
2 Homogeneous systems of linear equations with more variables than equations has nonzero solutions (Axler3.26)
2.1 Intuition
If you have more variables than equations, then some variables won't be restrained by the equations.
3 Inhomogeneous systems of linear equations with more equations than variables have no solution for some choice of constant terms
3.1 Intuition
If you have more equations than variables, then some choice of constant terms (some set of equations) will be to constraining.
4 methodology
Basically turn it into a linear map and then abuse the injective/surjective theorems