If you put 10 coins into your bank account and the balance ends up being +3 coins, how much balance did you have to start with?

Technically speaking, yes, you can say that "no solution is possible" since there is no such thing as an anti-coin (or even "half a coin," depending on how you define it) but life becomes immensely easier if you accept the concept of "negative coins" to represent debt. That way you can easily have a balance of -7 on your account and offset that by just adding money to the account.

A similar thing happens with imaginary numbers. Instead of representing things which flip between two states (positive and negative) they rotate between four states, and like negative numbers, that enables a lot of extremely important math.