What can we expect

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Section 1.1 What can we expect

In these first explorations, we aim to develop some intuition for the type of behavior we can expect to see when looking at solutions of systems of linear equations. The solution to a linear equation in three unknowns \(x\text{,}\) \(y\text{,}\) and \(z\) is a plane. Use \(3\times5\) cards as models for planes to study the solutions to systems of linear equations in three variables.

Activity 1.1.0.1.

Consider a system of two equations in three variables. Use two notecards to model the two planes. For each of the following questions, be prepared to explain your answer and/or give an example.

(a)

Is it possible that there are no solutions to a system of two equations in three unknowns?

possible
not possible

(b)

Is it possible that the solution to a system of two equations in three unknowns is a single point?

possible
not possible

(c)

If you are studying a system of two equations in three variables, what would be the most likely solution set?

the empty set (no solution)
exactly one solution
infinitely many solutions in the form of a line
infinitely many solutions in the form of a plane

Activity 1.1.0.2.

Consider a system of four equations in three uknowns. Use four notecards to model the four planes. For each of the following questions, be prepared to explain your answer and/or give an example.

(a)

Is it possible that the solution set for four equations in three variables forms a line?

not possible
possible

(b)

Is it possible that the solution set for four equations in three variables forms a plane?

not possible
possible

(c)

If you are studying a system of four equations in three variables, what would be the most likely solution set?

the empty set (no solution)
exactly one solution
infinitely many solutions in the form of a line
infinitely many solutions in the form of a plane

Activity 1.1.0.3.

Suppose we have 500 linear equations in 10 unknowns. The most likely solution set would be

the empty set (no solution).
exactly one solution.
infinitely many solutions.

Activity 1.1.0.4.

Suppose we have 10 linear equations in 500 unknowns. The most likely solution set would be

the empty set (no solution).
exactly one solution.
infinitely many solutions.

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