We began our explorations in What can we expect by noticing that the solution spaces of linear systems with more equations seem to be smaller. Let's reexamine this idea using what we know about pivot positions.
Remember that the solution space of a single linear equation in three variables is a plane. Can two planes ever intersect in a single point? What are the possible ways in which two planes can intersect? How can our understanding of pivot positions help answer these questions?
Suppose that a consistent linear system has more variables than equations. By considering the possible pivot positions, what can you say with certainty about the solution space?
If a linear system has many more equations than variables, why is it reasonable to expect the system to be inconsistent?
Austin, Understanding Linear Algebra, §1.4 Pivots and their influence on solution spaces, Exercise 5