Why do you use subtraction when you add a positive integer and a negative integer?

Because adding a negative is the same as subtracting a positive. In other words, minus a plus is he same as plus a minus. For example

5+ (-3) = 5-3 = +2

5 -(+3) = 5-3 = +2

Remember

minus a plus = = minus

plus a minus = minus

plus a plus = plus

minus a minus = plus

Because by definition, adding a negative integer is the same as subtracting its positive counterpart.