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7x+18y=208 18=7(2)+4 7=4(1)+3 4=3(1)+1 GCD(7,18)=1 1=4−3(1)=4−(7−4)=7(−1)+4(2) 7(−1)+(18(2)−7(4))=7(−5)+18(2) xo=−1040;;yo=416 I know the general solution it a combination of the particular solution and the original equation but I don't understand the different signs, so how do one get the complete solution in general?

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which signs?
The answer to this specific equation is xn=-1040+18n and yn=416-7n and the signs I mean is the pluses and minuses in that solution cause they seem to varie
- - = + + + = + + - = - - + = -

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But how, if the specific solution for x=-1040 and the y in the equation is 18 so then the answer is -1040+18n but what if y would have been -18. Is the solution 1040+18n then?
once u have a particular solution x0 and y0 for the equation \(ax+by=c\) then the general solution is given by \[ \large x=x_0+\frac{b}{d}t\qquad\text{and}\qquad y=y_0-\frac{a}{d}t \] or \[ \large x=x_0-\frac{b}{d}t\qquad\text{and}\qquad y=y_0+\frac{a}{d}t \]
notice that the signs are apposite. if x has the positive then y has the negative, and vice versa.
Thank, really useful! Just to be clear, the first one is ax+by=c and the second is ax-by=c, right?
By the way, whats the d for?
d=gcd(a,b) you have two possibilities regarding the sign. i wrote both possibilities.

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