Now work through those equations in the reverse order to get
So one solution to the equation 42x+55y=1 is x=–17, y= 13. You get the other solutions by adding a multiple of 55 to x, and subtracting the same multiple of 42 from y. Thus (you can see that the multiples of k cancel, to leave you with the result from the previous displayed equation).
Now all you have to do is to choose k so that –17+55k lies between 100 and 200.