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Under what conditions on a, b, c is it true that the equation
ax + by + cz = 1 has a solution?
What is a general method of finding a solution when one exists?
Are there any other restrictions to this question?
At the moment, you could just have x = 1/a, y=z=0, regardless of what a, b and c are. There are infinite other solutions.
Why did the vector cross the road?
It wanted to be normal.
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In order for the equation to have at least one solution, a, b, c must not be all 0. Conversely, if at least one of a, b, c is not 0, then there will be a solution (indeed infinitely many solutions).
To find a solution, note which of a, b, c is not zero. Say, suppose c ≠ 0. Then set x = y = 0 and so you get (0,0,1⁄c) as one possible solution. If b ≠ 0, you can set x = z = 0, and if If a ≠ 0, set y = z = 0.
Last edited by JaneFairfax (2007-09-17 23:04:01)
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Sorry, I guess I should have mentioned that the equation is written in the form:
ax + by + cz = gcd(a,b,c)