12.6 Solutions to Consistent and Inconsistent Problems
Inconsistent
The last line of the augmented matrix after EROs states that
0x1 + 0x2 ≠ -0.5.
These are parallel lines. No real numbers solve this system.
Consistent and underdetermined
The last line of the augmented matrix after EROs states that 0x1 + 0x2 = 0.
There are an infinite number of solutions along a line because equations express the same line.
Consistent and uniquely determined
A solution can be obtained.
This represents a single point where two lines intersect.
Consistent and underdetermined
The last line of the augmented matrix after EROs states that
0x1 + 0x2 + 0x3 = 0.
This represents an infinite number of solutions along a line where two planes intersect. The third plane is a linear combination of the other two planes, so it gives us no further information.
Consistent and uniquely determined
A solution can be obtained.
This represents a single point where three planes intersect.
Consistent and underdetermined
The last two lines of the augmented matrix after EROs states that
0x1 + 0x2 + 0x3 = 0.
This system has an infinite number of solutions in a plane. All three equations represent the same plane.
Inconsistent
The last line of the augmented matrix after EROs states that 0x2 + 0x3 ≠ -0.5
This represents parallel planes. No real numbers solve this system because no real numbered solution lies in the intersection of these three planes. This tells us that at least two of the planes must be parallel to one another. Note that they do not all three have to be parallel. Even if two planes intersect, if the third plane is parallel to one of the first two planes, there will be no point that lies in all three planes.
Those that are consistent and uniquely determined - c and e.
