[Systems of linear equations] $[\quad$ Case enumeration $\quad]$

Try to change a 25-ruble bill into eleven bills of 1, 3, and 5 rubles.
If we take 11 three-ruble bills, we get 33 rubles - 8 rubles more than needed. Replacing several three-ruble bills with one-ruble bills reduces the sum by 2 rubles for each replacement. Therefore, to reduce the sum by 8 rubles, we need to replace 4 three-ruble bills with 4 one-ruble bills: $7 \cdot 3 + 4 \cdot 1 = 25$.

## Answer

$25 = 4 \cdot \mathbf{1} + 7 \cdot \mathbf{3} = 5 \cdot \mathbf{1} + 5 \cdot \mathbf{3} + 1 \cdot 5 = 6 \cdot \mathbf{1} + 3 \cdot \mathbf{3} + 2 \cdot \mathbf{5} = 7 \cdot \mathbf{1} + 1 \cdot \mathbf{3} + 3 \cdot \mathbf{5}$.