## frx 3 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?

1. Muskan

which signs?

2. frx

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

3. Muskan

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

4. frx

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?

5. helder_edwin

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

6. helder_edwin

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

7. frx

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

8. frx

By the way, whats the d for?

9. helder_edwin

d=gcd(a,b) you have two possibilities regarding the sign. i wrote both possibilities.