I know what they are, I'm wondering if its possible to solve for
things without knowing them
William Beaty's answer is the correct one and I wish to expand on it some since this question frequently comes up.
When assigning a voltage or current variable for circuit analysis, one must choose a reference polarity for a voltage and a reference direction for a current, just as, when placing a voltmeter or ammeter in a circuit, one must choose an orientation for the leads. You cannot avoid choosing a reference polarity / direction.
Often, students learning the process are 'afraid' of choosing incorrectly but, rest assured, that's impossible. Indeed, two students can choose opposite reference polarities and directions and, assuming no mistakes, both will solve the circuit correctly. Yes, their answers will differ by a sign but both answers give the same information - the magnitude and the absolute polarities / directions.
So, what exactly does the reference polarity / direction imply?
For example, in the circuit diagram, we have the topmost terminal of the 6k resistor labelled positive. Does this mean that we 'believe' the topmost terminal is actually more positive? No!
It means that the voltage we calculate for this variable is the voltage we will measure if we place the red lead on the topmost terminal and the black lead on the bottommost terminal.
Obviously, if we reverse the reference polarity, the sign of the calculated answer changes just as when we reverse the leads of the voltmeter, the sign of the measured voltage changes.
But the choice of reference polarity cannot affect the absolute polarity of the voltage.
Thus, if we place the leads of the voltmeter across a circuit element and measure a positive voltage, we know that the terminal connected to the red lead is more positive than the other terminal.
Conversely, if we measure a negative voltage, we know that the terminal connected to the red lead is less positive than the other terminal, i.e., the terminal connected to the black lead is more positive.
To summarize, by picking a reference polarity / direction (which we must do), we're not 'guessing' at that absolute polarity / direction; the calculated or measured result will tell us that.