Activity1.1.2

This activity begins with equations having three unknowns. In this case, we know that the solutions of a single equation form a plane. If it helps with visualization, consider using \(3\times5\) inch index cards to represent planes.

  1. Is it possible that there are no solutions to two linear equations in three unknowns? Either sketch an example or give a reason why it can't happen.

  2. Is it possible that there is exactly one solution to two linear equations in three unknowns? Either sketch an example or give a reason why it can't happen.

  3. Is it possible that the solutions to four equations in three unknowns form a line? Either sketch an example or give a reason why it can't happen.

  4. What would you usually expect for the set of solutions to four equations in three unknowns?

  5. Suppose we have 500 linear equations in 10 unknowns. What would be a reasonable guess for which of the three possibilities for the set of solutions holds?

  6. Suppose we have 10 linear equations in 500 unknowns. What would be a reasonable guess for which of the three possibilities for the set of solutions holds?

in-context