On Sun, 27 Jan 2008 21:35:45 -0800 (PST), Citizen
<FlammesSombres@[EMAIL PROTECTED]
> wrote:
>Why is it impossible for a system of equations to have exactly 2
>solutions?
>
>Thank you very much for your help!
It is possible for a system of equations to have exactly two solutions.
However, from the subject line, I infer that the question you actually
meant to ask is "Why is it impossible for a system of linear equations
to have exactly two solutions?"
Consider what it means to be a linear equation. The term comes from the
graph of the solution of the equation being a straight line on the
coordinate plane. A system of such equations is graphed with several
straight lines and the system has its solution(s) where the lines all
cross. Consider a simple system of just two equations. When you graph
them as lines, there three possibilities: (1) the lines run parallel, in
which case the system has no solution, (2) the lines overlap (i.e., they
are the same line), in which case the system has infinitely many
solutions (every point on the line is a solution of the system), or (3)
the lines intersect at one point, in which case the system has exactly
one solution. In order to cross a second time, one line would have to
curve back towards the other again, which would make it not a straight
line.
There are other ways of explaining it, but I think for most, thinking in
terms of the graphs of the equations makes it most intuitively
accessible. This is not, of course, a proof, but you did not ask for
one, so I assumed you were just trying to reach an intuitive
understanding of something you had been taught as a bare fact. If this
has not been helpful, perhaps you could elaborate on your question.
--
R. Dan Henry
danhenry@[EMAIL PROTECTED]
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