Question

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?

Answer 1

which signs?

Answer 2

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

Answer 3

- - = +
+ + = +
+ - = -
- + = -

Answer 4

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?

Answer 5

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
\[ \large x=x_0-\frac{b}{d}t\qquad\text{and}\qquad y=y_0+\frac{a}{d}t \]

Answer 6

notice that the signs are apposite. if x has the positive then y has the negative, and vice versa.

Answer 7

Thank, really useful! Just to be clear, the first one is ax+by=c and the second is ax-by=c, right?

Answer 8

By the way, whats the d for?

Answer 9

d=gcd(a,b)

you have two possibilities regarding the sign. i wrote both possibilities.