Pressure at the same depth in connected fluid systems

[samsanch0803]

TL;DR

I was solving a hydrostatic physics exercise and I was going to choose the middle line, the one that does not belong to the interface, as my reference point to equalize the pressures since they are at the same height, but I realize that the calculation there does not give what it should give.
So here my question is why the pressures on the purple dots are not the same as what happens on the red dots if they are at the same height?

 

[Tazerfish]

For any static fluid volume that you look at, the forces balance.
The pressure force between the bottom and top surface, i.e. the buoyancy, cancel out the gravitational force.
If it's all the same fluid then the pressure at every height is the same.

Here however, you get a different pressure at the purple points because the mercury is denser than the water. Therefore the gravitational force is stronger and in order to be balanced, the pressure force must be stronger too.

Does that make sense?

 

[kuruman]

On the right is a figure showing the variation of pressure with distance $y$ from an arbitrary horizontal level O. The green liquid has higher mass density than the brown fluid, $\rho_{\text{green}}>\rho_{\text{brown}}.$
Note that
1. For $0<y<A$ the pressure is atmospheric everywhere.
2. For $A<y<B$ the pressure is atmospheric in the left tube while it increases linearly above atmospheric in the right tube. The slope of the straight line is $s=\rho_{\text{brown}}~g.$
3. For $B<y<D$ the pressure increases linearly in both tubes. The green slope is higher and the green line catches up with the brown line at point D which is the interface between the fluids. However at any point C (red dashed line) at the same height the pressures are not equal.

 

[Tazerfish]

That's a beautiful post kuruman.!
I find the axes a bit unintuitive; my brain expects them to be oriented the same way as the diagram to the left—height y on the y axis and p(y) on the x-axis, but perhaps that'd be even more confusing.

Other than that though...
A picture says more than 1000 words.

 

[Lnewqban]

... So here my question is why the pressures on the purple dots are not the same as what happens on the red dots if they are at the same height?

Welcome, @samsanch0803 !

Please consider that the hydrostatic pressure depends on two factors: density and height.

Same height means same pressure only if density is also the same.

As density decreases, the height of fluid column must increase, if the same hydrostatic pressure is expected.

As the pressure at both free surfaces must be equal to the atmospheric one, and the pressure along any horizontal line crossing both columns of the same type of fluid (mercury in this case) must be equal to each other; the variation or gradient of pressures should be proportional between two types of fluids of different densities (mercury and water in this case as well).

 

[A.T.]

TL;DR Summary: I was solving a hydrostatic physics exercise ...

The key to understanding hydrostatics is the static part:

The forces acting on any subvolume of fluid must balance. From this it follows that columns of different density but same height require different pressure at the bottom to support them against gravity.

 

[kuruman]

The key to understanding hydrostatics is the static part:

The forces acting on any subvolume of fluid must balance. From this it follows that columns of different density but same height require different pressure at the bottom to support them against gravity.

All that is true but does not address OP's question. OP's drawing shows two horizontal lines P3-P4 (red) and P1-P2 (purple). The fluids above each line have different densities and different heights to the surface. The OP asks how come that the pressures on each side along P3-P4 are the same while not the same along P1-P2?