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Question 169956: Do two linear equations always have a solution? if not why? if yes, under what comditions?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! First, I have to presume you are talking about two-variable linear equations. Second, by 'solution,' I presume you mean a single ordered pair that satisfies both equations. Given that the answer to your first question is no.
There are three possibilities for any given pair of two-variable linear equations:
1. There is a unique ordered pair that satisfies both equations. This is analogous to the point of intersection of the two lines in the coordinate plane represented by the two equations. The point of intersection is described by the ordered pair that is the solution of the two equations.
2. There is no ordered pair that satisfies both equations. This is analogous to the two lines represented by the two equations being parallel. The two lines never intersect, so there is no ordered pair that fits both equations.
3. There are an infinite number of ordered pairs that satisfy both equations. This is analogous to the two lines represented by the two equations being the same line. The two lines intersect in ALL points on either line, and there are an infinite number of points in any line, hence an infinite number of points of intersection. Said another way, if an order pair satisfies one of the equations, it will always satisfy the other equation as well.
Hope this helps,
John
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