What is the node 1 voltage in the circuit of the figure?

Steps to Solve

1. Identify the Circuit Configuration
   The circuit includes multiple resistors and current sources. Label the nodes and identify known voltages and currents.

2. Apply KCL at Node 1
   Using Kirchhoff's Current Law (KCL), set up the equation for the current flowing into and out of Node 1.
   Let the voltage at Node 1 be ( V_1 ). The equation can be expressed as:
   $$ \frac{V_1 - V_{2}}{6} + \frac{V_1}{2} + 3 = 0 $$ Here, ( V_{2} ) is the voltage of the reference node, which is usually taken as 0V.

3. Substitute Known Values
   Substitute the value ( V_{2} = 0 ) V into the equation:
   $$ \frac{V_1}{6} + \frac{V_1}{2} + 3 = 0 $$

4. Combine Terms
   To solve the equation, convert all terms to a common denominator (in this case, 6):
   $$ \frac{V_1}{6} + \frac{3V_1}{6} + 3 = 0 $$ Combining gives: $$ \frac{4V_1}{6} + 3 = 0 $$

5. Isolate ( V_1 )
   Rearrange the equation to solve for ( V_1 ):
   $$ \frac{4V_1}{6} = -3 $$
   Multiply each side by ( 6 ): $$ 4V_1 = -18 $$ Then divide by 4: $$ V_1 = -\frac{18}{4} = -4.5 , \text{V} $$

6. Checking Values
   Round or adjust the value if necessary based on choices:
   A recent approximation could lead you to the correct answer from the provided options.