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    Suppose I need to find integer solutions to Ax+By=C. I know that there will also be solutions for every A(x+c)+B(x+d)=14 such that Ac+Bd = 0.

    Suppose I already know the values of x and y. Is there a quick way to find values for c and d without having to solve another equation?
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    (Original post by MR1999)
    Suppose I need to find integer solutions to Ax+By=C. I know that there will also be solutions for every A(x+c)+B(x+d)=14 such that Ac+Bd = 0.

    Suppose I already know the values of x and y. Is there a quick way to find values for c and d without having to solve another equation?
    I'm puzzled, don't you just read solutions off from Ac+Bd = 0? I'm assuming that as you've called this a Diophantine problem, A and B are integers.
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    (Original post by MR1999)
    Suppose I need to find integer solutions to Ax+By=C. I know that there will also be solutions for every A(x+c)+B(x+d)=14 such that Ac+Bd = 0.

    Suppose I already know the values of x and y. Is there a quick way to find values for c and d without having to solve another equation?
    (Original post by Gregorius)
    I'm puzzled, don't you just read solutions off from Ac+Bd = 0? I'm assuming that as you've called this a Diophantine problem, A and B are integers.
    I think the question has been badly paraphrased (in particular, what has 14 got to do with the problem as stated...?)

    But I think what the OP actually wants to know is (with me giving concrete examples):

    Suppose we have a simple Diophantine equation such as 18x+15y = 15.

    It's obvious that x = 0, y = 1 is a solution, but the general form x = 5n, y=1-6n is less obvious.

    But (to the OP), it's still pretty straightforward: it's clear that 18(15) + 15(-18) = 0, (which would give x = 15n, y = 1-18n). For full generality we divide by the highest common factor of (18, 15), which is 3.
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    (Original post by DFranklin)
    I think the question has been badly paraphrased (in particular, what has 14 got to do with the problem as stated...?)

    But I think what the OP actually wants to know is (with me giving concrete examples):

    Suppose we have a simple Diophantine equation such as 18x+15y = 15.

    It's obvious that x = 0, y = 1 is a solution, but the general form x = 5n, y=1-6n is less obvious.

    But (to the OP), it's still pretty straightforward: it's clear that 18(15) + 15(-18) = 0, (which would give x = 15n, y = 1-18n). For full generality we divide by the highest common factor of (18, 15), which is 3.
    Ah, thanks!
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    (Original post by DFranklin)
    I think the question has been badly paraphrased (in particular, what has 14 got to do with the problem as stated...?)

    But I think what the OP actually wants to know is (with me giving concrete examples):

    Suppose we have a simple Diophantine equation such as 18x+15y = 15.

    It's obvious that x = 0, y = 1 is a solution, but the general form x = 5n, y=1-6n is less obvious.

    But (to the OP), it's still pretty straightforward: it's clear that 18(15) + 15(-18) = 0, (which would give x = 15n, y = 1-18n). For full generality we divide by the highest common factor of (18, 15), which is 3.
    Yeah I guess the question was very badly worded. The given equation was 826x+350y=14 but I wanted to make it more general.

    So given the solution pair (-11,26), we can say that 826(-11+25n) +350(26-59n) = 14, for n an integer?
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    (Original post by MR1999)
    Yeah I guess the question was very badly worded. The given equation was 826x+350y=14 but I wanted to make it more general.

    So given the solution pair (-11,26), we can say that 826(-11+25n) +350(26-59n) = 14, for n an integer?
    Looks right.
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    I'm not sure but this may be of help for you:

    https://www.youtube.com/watch?v=TIk3ujphMfk
 
 
 
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