Finding node voltage of a “Y” resistor network

I am trying to find the voltage at NODE1. My initial approach was to find the current starting from the $15$V source then going through $R_1$ and $R_3$ to ground. Then calculating the current from $14$V source through $R_2$ and $R_3$ to ground. Then adding the two currents and using Ohm’s law with total current and $R_3$ to find voltage from NODE1 to ground. This seems plausible to me only because of KCL (total current entering a node must equal current exiting node). However my answer is incorrect as I have proved both in SPICE and on breadboard.

What am I missing? I would appreciate any suggestions that don’t totally give me the answer.

^{simulate this circuit – Schematic created using CircuitLab}