Systems of Linear Equations

Part 6: Algebraic Interpretations of the Geometry

1. Give an example of a system which represents three lines that intersect in a single point.
2. Give an example of a system which represents three lines and has no solution.
3. A system of three equations in three unknowns represents a system of three planes. If the three planes coincide, there will be infinitely many solutions. In what other way(s) can we have infinitely many solutions? For each way, write down an example of such a system.
4. The pictures below show two possible configurations of three planes with no common intersection.
   • On the left, no two of the planes are parallel, but each pair intersects in a line, and the three lines are parallel.

   • On the right, two of the three planes are parallel, and the third intersects them in two lines (which are also parallel).

   Which of these configurations is similar to the configuration for an inconsistent system that you examined in Step 6 of Part 5? For the other configuration, write down a system of equations the picture could represent.

5. If a system represents two planes that intersect in a line, how many equations are in the system? Is the system consistent or inconsistent? Can you say anthing else about the system?