Finding node voltage of a "Y" resistor network
The easiest way to do this is to use superposition. Set one source to \$0\$ (i.e. shorted) and find the node voltage due to the other source, then set the other source to \$0\$ and find the node voltage due to the first source. Add the two contributions for the node voltage due to both sources.

For the case with the \$14\$V source set to \$0\$ (finding the contribution of the \$15\$V source) the circuit looks like this:







simulate this circuit &ndash; Schematic created using CircuitLab

\$R_2\$ and \$R_3\$ are in parallel since \$V_2\$ is shorted, and \$V_x\$ can be found using a simple voltage divider:

$$V_x = \frac{R_2||R_3}{R_2||R_3 + R_1}15\text{V}$$

It's a similar process for finding \$V_y\$, the contribution of the \$14\$V source to the node voltage (with the \$15\$V source set to \$0\$).

Then by superposition $$V_{\text{NODE}1} = V_x + V_y$$