Question

a) use the node voltage method to find v0 for the circuit

Explanation:

Step1: Define node voltages

Let the reference node be at the bottom - most node. Let the voltage at the left - hand node with respect to the reference node be $v_{\Delta}$ and the voltage at the right - hand node be $v_o$.

Step2: Write KCL equation at the left - hand node

The currents entering the left - hand node:
The current through the $500\Omega$ resistor is $\frac{50 - v_{\Delta}}{500}$, the current through the $1k\Omega$ resistor is $\frac{v_{\Delta}}{1000}$, and the current through the $2k\Omega$ resistor is $\frac{v_{\Delta}-v_o}{2000}$.
By Kirchhoff's Current Law (KCL), $\frac{50 - v_{\Delta}}{500}=\frac{v_{\Delta}}{1000}+\frac{v_{\Delta}-v_o}{2000}$.
Multiply through by 2000 to clear the fractions:
$4(50 - v_{\Delta}) = 2v_{\Delta}+(v_{\Delta}-v_o)$
$200-4v_{\Delta}=2v_{\Delta}+v_{\Delta}-v_o$
$200-4v_{\Delta}=3v_{\Delta}-v_o$
$v_o = 7v_{\Delta}-200$.

Step3: Write KCL equation at the right - hand node

The current through the $2k\Omega$ resistor is $\frac{v_{\Delta}-v_o}{2000}$, and the current through the $200\Omega$ resistor is $\frac{v_o}{200}$, and the current of the dependent current source is $\frac{v_{\Delta}}{750}$.
By KCL, $\frac{v_{\Delta}-v_o}{2000}+\frac{v_{\Delta}}{750}=\frac{v_o}{200}$.
Multiply through by 6000 to clear the fractions:
$3(v_{\Delta}-v_o)+8v_{\Delta}=30v_o$
$3v_{\Delta}-3v_o + 8v_{\Delta}=30v_o$
$11v_{\Delta}=33v_o$
$v_{\Delta}=3v_o$.

Step4: Substitute $v_{\Delta}=3v_o$ into $v_o = 7v_{\Delta}-200$

$v_o=7(3v_o)-200$
$v_o = 21v_o-200$
$20v_o=200$
$v_o = 10\ V$.

Answer:

$10\ V$