I'm not sure why I didn't see this connection before but now I've had a chance to think about this it is obvious how to generate equations, with integer co-efficient, passing through the same point.
Take the point (- 2, 6). I just have to set up the co-efficient's for the x and y terms and then find out what they sum to.
For instance I want one equation to involve 2x and 3y and the other to involve 3x and 2y.
So I write:
2x + 3y = 2(-2) + 3(6) = 14.
Giving me the equation:
2x + 3y = 14.
The second one is then:
3x + 2y = 3(-2) + 2(6) = 6
Giving me the second equation:
3x + 2y = 6.
I can vary the signs involved and can produce an infinite variety of simultaneous equations. These can vary in difficulty depending on the skills I wish to develop in a student.
This is connected to Diophantine equations and I can't remember most of the stuff I've learnt about these. Still this is a practical application I never thought of.
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