Question: Given:

```
width: 1920
height: 1080
total pixels: width * height = 2073600
aspect ratio: 1920 / 1080 ~= 1.8

```

How do I calculate a new resolution (width and height) with the same aspect ratio but a new pixel count? In the other words one that satisifes the following conditions:

```
width * height = 10000
width / height = 1.8

```

There are infintely many solutions to that equation system however and I want the solution that most resembles a screen resolution.
I suppose that means the solution with the minimum distance abs(widht - height), but I'm not really sure.

Answer: 
Note that you have two variables, $width$ and $height$, and two equations. There is only one solution.

```
height = aspect_ratio * width
width / height = aspect_ratio

width * height = total_pixels
aspect_ratio * width² = total_pixels

```

Solve to $width$ and you have $height$.
You have two equations in two unknowns, so there should generally be a unique solution. In your example, you can find width^2=18000, so width is about 134. Then height is about 134/1.8 or about 74. Now the problem is that integer division and square roots are not exact, so you need to specify what you are willing to compromise. 74\*134=9916, which is very close to 10000-can you live with that? If not, explore numbers nearby and choose your favorite.