When do systems have integer solutions?

Question

Suppose you have a system of the form:

ax+by=e and cx+dy=f, where a,b,c,d,e, and f are all integers...and it has only one solution. Then in each case calculate ad-bc. When would the system as described above have a solution with integer coordinates vs. a solution with coordinates that are not integers? How would the calculation of ad-bc help me figure this out?

Answer 1

the solution is: x = (ed-fb)/(ad-bc) and y = (af-ec)/(ad-bc)

ad-bc is the denominator in both solutions

it can't = 0

to be integers for x and y, ad-bc must divide into both af-ec and ed-fb evenly with no remainder

af-ec, ed-fb, and ad-bc are all integers, but an integer divided by an integer may or may not be an integer