What are the correct equations for loops 1 and 2?; Using Kirchhoff's Rules

We do not know the values for any of the components, so we will only be using Kirchhoff's Voltage Rule based off of conservation of energy: the sum of all Voltages in a loop must equal zero.

For Loop 1, we'll start at Voltage source 1 going CLOCKWISE. This battery is oriented in the same direction so we add its voltage. Next, we come across a resistor. Its voltage is the product of its resistance and its current. The current is going in the same direction as we are, so we subtract its voltage. Next we cross another voltage source. It is oriented in the opposite direction as us so we subtract its voltage. Finally we cross a resistor where the current is going in the same direction as us, so we subtract its voltage.

For Loop 2, we'll start at Voltage source 2 going CLOCKWISE. This battery is oriented in the same direction so we add its voltage. Next, we come across a resistor. Its voltage is the product of its resistance and its current. The current is going in the opposite direction, so we add its voltage. Next we cross another voltage source. It is oriented in the opposite direction as us so we subtract its voltage. Finally we cross a resistor where the current is going in the opposite direction as us, so we add its voltage.