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#1 2007-09-17 15:35:52

Marko
Guest

Question

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?

#2 2007-09-17 21:44:54

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Question

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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#3 2007-09-17 22:49:40

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Question

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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#4 2007-09-19 10:41:52

Marko
Guest

Re: Question

Sorry, I guess I should have mentioned that the equation is written in the form:
ax + by + cz = gcd(a,b,c)

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