Polarity in Circuit Analysis – Importance of Polarity in Calculating Current Through a Resistor

Personally, I find it "singing to my mind" better when I take a moment to redraw a schematic.

^{simulate this circuit – Schematic created using CircuitLab}

Do you see what I've done? It's the same circuit on the right. But I've removed some of the confusing wiring. Also, it's now perfectly clear that one of these nodes (wires) is at \$20\:\textrm{V}\$, too.

Now it is very easy to see what is happening. There is \$20\:\textrm{V}\$ across \$R_2\$. So you know the current in \$R_2\$. Also, there is \$5\:\textrm{V}\$ across \$R_1\$. So you know the current in \$R_1\$. Whatever is left over must be the current in \$V_2\$.

So,

$$\begin{align*} I_{V_2} &= I_{R_2} - I_{R_1}\\\\ &= \frac{20\:\textrm{V}}{R_2} - \frac{15\:\textrm{V}-20\:\textrm{V}}{R_2}\\\\ &= \frac{20\:\textrm{V}}{300\:\textrm{k}\Omega} - \frac{15\:\textrm{V}-20\:\textrm{V}}{19\:\textrm{k}\Omega}\\\\ &= 329.824561\:\mu\textrm{A} \end{align*}$$

I prefer simplifying my understanding of a problem before choosing a method for solving it by hand.

Of course, there are rigorous methods you must also learn to apply "by rote." Computer software in electronic simulators do this all day long. But when you are first learning these things, I think it helps a lot to try and redraw things to help "see" them better.

I get the following four equations:

$$\begin{align*} 0\:\text{V}+24\:\text{V}-5\:\Omega\cdot\left(I_1-I_3\right)-V_{4\text{A}}-3\:\Omega\cdot 4\:\text{A}&=0\:\text{A} \\\\ 0\:\text{V}+3\:\Omega\cdot 4\:\text{A}+V_{4\text{A}}-10\:\Omega\cdot\left(I_2-I_3\right)-20\:\Omega\cdot I_2&=0\:\text{A} \\\\ 0\:\text{V}-5\:\Omega\cdot\left(I_3-I_1\right)-5\:\Omega\cdot I_3-10\:\Omega\cdot\left(I_3-I_2\right)&=0\:\text{A} \\\\ I_1-I_2&=4\:\text{A} \end{align*}$$

where \$V_{4\text{A}}\$ is the voltage across the current source.

I used the + sign on the top node and - on the bottom node of the current source. As shown below:

No need for any super- anything.

These solve out (using sympy):

var('i1 i2 i3 v4a')
eq1 = Eq( 0 + 24 - 5*(i1-i3) - v4a - 3*4, 0 )
eq2 = Eq( 0 + 3*4 + v4a - 10*(i2-i3) - 20*i2, 0 )
eq3 = Eq( 0 - 5*(i3-i1) - 5*i3 - 10*(i3-i2), 0 )
eq4 = Eq( i1 - i2, 4 )
solve( [ eq1, eq2, eq3, eq4 ], [ i1, i2, i3, v4a ] )

{i1: 24/5, i2: 4/5, i3: 8/5, v4a: -4}

Check your work against mine to see if you agree. Then check all this against the stated solution.

Toss the textbook into the trash can. And with it, the 8th edition of "Electronic Principles" by Malvino & Bates. And probably dozens of other terrible books on the topic of electronics. I've no idea how they get so bad, but they do. Even after 8 editions! The world is flush with garbage. I'm guessing it has to do, in part, with grad students. But I'm not sure.