Think like this, we can find the values of variables if the number of equations and number of variables is same.
rather than give a "formal" answer i'll give an "informal one". one can think of the dimension of a vector space as quantifying its "size" in some sense. a system of two equations in 3 unknowns is equivalent to a 2x3 matrix, which takes your inputs (the 3 unknowns) and spits out two numbers (the two equations evaluated at the inputs).
so we have a mapping from a three-dimensional space to a two-dimensional one. this can't happen without at least one dimension "collapsing" down to 0. the dimension(s) that collapses is some line in the three-dimension space that gets mapped to a single point (the origin of the two-dimensional space). this is a many-to-one operation, which is NOT uniquely reversible.
all of this can be formalized by using the concepts of rank, linear independence and spanning set, but the "idea" is what i just said above.
The rank of augmented matrix A:B and A have to be same(=r say) for solution to exist, if r=n unique solution, r<n, infinitely many solutions . In case of 2 equations and 3 variables the rank of augmented matrix and matrix A can be at the most 2 wich is less than 3, so infinitely many solutions exist wich can be found out by putting one unknown equal to a parameter and finding other unknowns in terms of that parameter.