A $45$-gallon bathtub is to be filled with water that is exactly 105$^{\circ}$F. Unfortunately, the two sources of water available are 120$^{\circ}$F and 60$^{\circ}$F. When mixed, the temperature will be a weighted average based on the amount of each water source in the mix. How much of each should be used to fill the tub as specified?

Amount of 120$^{\circ}$F water: gallons
Amount of 60$^{\circ}$F water: gallons