This exercise has nothing to do with inverse operations.
Let L represent the number of lawns Tyler mows, and D the number of dogs he walks.
If he earns $20 for each lawn mown, then mowing L lawns earns him $20L. For example, if he mows 1 lawn, he gets $20. If he mows 2 lawns, he earns $40. And so on.
The same could be said about dog-walking. If he earns $15 for each walk, then walking D dogs is worth $15D. For example, if he walks 1 dog, he gets $15; for 2 dogs, $30; and so on.
If after a week he has earned a total of $260, then
20L + 15D = 260
which can optionally be simplified a bit by dividing through both sides by 5 to get
4L + 3D = 52
Since you mentioned "inverse", I suspect you interpreted the question as asking how many lawns Tyler mowed and how many dogs he walked. This is impossible, you cannot determine exactly how many lawns or dogs were involved. But there are only 4 ways this can happen, since L and D must be counting numbers.
D = 4, L = 10
D = 8, L = 7
D = 12, L = 4
D = 16, L = 1