Simple Node voltage method problem

[DWill]

Homework Statement

Use the node voltage method to find v in the circuit shown.

(picture of circuit attached)

Homework Equations

The Attempt at a Solution

v1 + (v1 - v2)/8 = 4.5
(v2 - v1)/8 + (v2)/12 + (v2 - 30)/4 = 0

Solving for v1 and v2 I get:
v1 = 6 V
v2 = 18 V

I got the voltage at node 1 and 2 (along the top of the circuit), but I'm puzzled as to how to get v? Thanks for your help.

 

[rootX]

Can you see v is equal to the voltage across the 2 ohm resistor and the 12 ohm resistor?

You know the 12 ohms voltage, so find the voltage across the 2 ohms to find v.

 

[vk6kro]

Homework Statement

Use the node voltage method to find v in the circuit shown.

(picture of circuit attached)

Homework Equations

The Attempt at a Solution

v1 + (v1 - v2)/8 = 4.5
(v2 - v1)/8 + (v2)/12 + (v2 - 30)/4 = 0

Solving for v1 and v2 I get:
v1 = 6 V
v2 = 18 V

I got the voltage at node 1 and 2 (along the top of the circuit), but I'm puzzled as to how to get v? Thanks for your help.

You found the voltages at the two node points, so you know the difference between them. So you know the current in the 6 and 2 ohm resistors by Ohms Law. If you work out the voltage across one of them, and add it to the nearest node point, you can get the voltage there relative to the bottom rail.