Question

Can a system of two linear equations in three variables like(below) be consistent, inconsistent, and dependant, like a two linear equation system with two variables? -2x+3y-z=-1 x-2y+z=3 What about a system of three equations with three variables?

Thanks

Answer 1

Yes

Explanation:

The simple answer is yes.

If you have two consistent equations and they are linearly independent, then you will have to assign arbitrary values to one of the variables. If you have two consistent equations and they are linearly dependent, then you will have to assign arbitrary values to two of the variables, both cases lead to an infinite number of solutions. If they inconsistent, then obviously there is no solution.

Graphically two consistent linearly independent equations will form 2 planes in `RR^3` that intersect and all solutions will lie on the line of intersection leading to an infinite number of solutions.

Two consistent linearly dependent equations will just be a single line in `RR^3` and all solutions will lie on this line, leading to an infinite number of solutions.

If the two equations are inconsistent then you will have 2 parallel lines in `RR^3`, and no point of intersection, hence no solutions.