Can a system of two linear equations have exactly two solutions?
2 Answers
No. A system of linear equations in two variables may have zero, one, or infinitely many solutions.
Explanation:
We can think about the geometry.
A linear equation has a graph the as a (straight) line.
Given a system of two linear equations, there are three possibiities.
The possibilities are:
two parallel lines
e.g.
two (distinct) intersecting lines -- that is, two different lines that meet at a point. Two straight lines cannot meet in two points. Two points determine one and only one line (not two lines).
e.g.
The equations have the same line as their graphs
If two lines do have two points in common, then "the lines coincide" (which really means "there is only one line".) In thiis case we say "the two lines are the same".
This odd phrase is not unique to mathematics. I have had students in the past who could tell there friends in ordinary English, "Two of my professors are the same person." Which, of course, really means there is only one professor but there are two descriptions of that teacher -- my math teacher and my philosophy teacher.
e.g.: (trivial)
Still trivial:
Not so trivial:
Under normal circumstances a system of two linear equations can have
Explanation:
Linear equations represent lines (or planes, etc., but essentially we are interested in lines for this question).
If two lines are coincident (i.e. the same line), then they intersect at all points along the line - that is, infinitely many points and hence infinitely many solutions.
If two lines are parallel (and non-coincident) then they do not intersect and there is no solution.
If two lines are coplanar and non-parallel, then they will intersect in exactly one point. That is exactly one solution. If they are not coplanar, they will have no intersection and therefore no solution.
Exceptions
If the coordinates range over something other than the Real numbers then it is possible for such a system to have exactly two solutions. For example,
