You may think you need to use loop equations to solve the problem, but if you look closely at the second loop you will see that the current through R1 is defined by the 2mA current source. Therefore, the voltage across R1 can be found simply by Ohms law: V_{R1} = I_{R1} × R1 = 2mA × 1kΩ = 2V.

Sure, if you wanted to calculate all the node voltages with respect to ground then you would need to use either loop equations or network reduction, but you don’t need that to answer the question as stated.

The moral of the exercise is to always look for the simplest solution first, and don’t make the problem harder than it really is.