it appears to me that the current generated by 35 V and 2vx will
collide each other
It may be that you are assuming that a voltage source, whether independent or controlled, must source current, i.e., supply power to the circuit.
But, at least in ideal circuit theory, there's nothing "wrong" with a voltage source sinking current, i.e., receiving power from the circuit.
For a real world example, consider that, when a battery is being charged, the current is in the opposite direction than when the battery is being discharged.
I would like to know how the current flows across 5 Ω resistor.
If you're planning to be an EE, don't write or say things like "current across"; current is through, voltage is across.
Now, this circuit is very easy to solve. There are two unknowns so you need two independent equations.
For the 1st, write a KVL equation clockwise 'round the loop:
$$35V = v_x + 2v_x - v_o \rightarrow 3v_x = 35V + v_o$$
Now, you need one more independent equation. Can you find one?
Let's call the 4 Ohm resistor R1.
We can express the current through R1 (in the left-to-right direction) as
\$I(\mathrm{R1}) = \frac{1}{\mathrm{R1}}\left(V_1 - \left(V_3 - 8V_b\right)\right)\$,
which is just what you've already done.
Now the only thing I think you're missing is that Vb can be expressed in terms of V2 and V3, so now
\$I(\mathrm{R1}) = \frac{1}{\mathrm{R1}}\left(V_1 - \left(V_3 - 8\left(V_2-V_3\right)\right)\right)\$,
which of course you can simplify further using the usual rules.
I don't know who Mark Scheme is, but I believe he made a sign error in his solution.
Best Answer
A current source can certainly have a voltage across it. If the voltage across a current source is zero, then it is not delivering or absorbing any power. However, if the voltage across the source is not zero, then it is either sourcing or sinking power into the rest of the circuit.