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frx
 2 years ago
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?
frx
 2 years ago
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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frx
 2 years ago
Best ResponseYou've already chosen the best response.0The answer to this specific equation is xn=1040+18n and yn=4167n and the signs I mean is the pluses and minuses in that solution cause they seem to varie

Muskan
 2 years ago
Best ResponseYou've already chosen the best response.1  = + + + = + +  =   + = 

frx
 2 years ago
Best ResponseYou've already chosen the best response.0But 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?

helder_edwin
 2 years ago
Best ResponseYou've already chosen the best response.1once 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 \[ \large x=x_0\frac{b}{d}t\qquad\text{and}\qquad y=y_0+\frac{a}{d}t \]

helder_edwin
 2 years ago
Best ResponseYou've already chosen the best response.1notice that the signs are apposite. if x has the positive then y has the negative, and vice versa.

frx
 2 years ago
Best ResponseYou've already chosen the best response.0Thank, really useful! Just to be clear, the first one is ax+by=c and the second is axby=c, right?

frx
 2 years ago
Best ResponseYou've already chosen the best response.0By the way, whats the d for?

helder_edwin
 2 years ago
Best ResponseYou've already chosen the best response.1d=gcd(a,b) you have two possibilities regarding the sign. i wrote both possibilities.
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