Here's a simple illustration of Coin Change Problem for two items, which can easily be represented in a two-dimensional coordinate system.
Given target sum, say 7, we need to determine coefficients for the two items, say 2 and 3, so that .
This can be done by fixing possible y coordinates (for 3) and determining corresponding x coordinates (for 2). Doing so we see that
- fixing earlier coordinates narrows down later choices
- not all earlier fixings can yield later possibilities
This is also related to Knapsack problem, which attains maximization bounded by a second set of coefficients.