Question 175595: Writing In a system of three linear equations with three variables,
the number of solutions depends on how the planes defined by the
equations intersect. List the different numbers of solutions that are
possible, and explain when each occurs.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! You could have the following situations:
1. One element in the solution set consisting of an ordered triple. This occurs when all three lines intersect in a single point.
The lines could all lie in the same plane, or
Two of the lines could be in one plane while the third line is in a different plane, in which case the solution set to the system would be an ordered triple representing a point on the line of intersection of the two planes. The two lines that lie in the same plane could be the same line.
All three lines lie in different planes, in which case the point of intersection of the lines and the point of intersection of the planes is the same point.
2. Infinite solutions. All three lines are the same line and any ordered triple that satisfied one of the equations would satisfy the other two.
3. No solution. The three lines do not intersect in the same point or do not intersect at all.
The following situations would have an empty solution set:
Three parallel lines in the same plane.
Three lines possibly, but not necessarily co-planar, such that any pair intersects, but the three possible pairs of lines intersect at different points.
At least one of the three lines, non-coplanar to the other two lines, lies in a plane parallel to a plane containing one or both of the other lines. Such a line is called a 'skew' line.
Two equations representing the same line and a third line parallel or skew to this line.
There may be others, but you get the idea.
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