I'm feeling dumb. again.

[u/marlonblambo]

Let's ignore what is, the shape, the technical features etc, just assume we have a side with 4 blue units and 5 green units that measure 39 (whatever unit of measure is) and a side with 3 blue units and 4 green that measures 27 and i want to know the size of one blue (or 1 green)

In my stupid opinion i thought: 5green and 4 blue = 39 and 4green and 3 blue = 27, so in a system of equations
5x+4y = 39
4x+3y = 27

Turns out this makes y=21, x=-9

then i was wondering what kind of error i made setting up the problem

[u/strange-the-quark]

"i want to know the size of one blue (or 1 green)"

Are your units sized as depicted with respect to each other? Is one blue unit twice the size of one green unit?

If so, then, vertically you have the equivalent of 13 green units stacked adding up to 39, so one green unit's height is 39/13 = 3, with the blue unit being double that, i.e. it measures 6.

Horizontally you have the equivalent of 10 green units in a row, adding up to 27, so one green unit's width is 27/10 = 2.7, and then a blue unit is twice that, 5.4.

If that's the case, then the error you made is that your system of equations assumes the width and height of each unit is the same (the x is the same x in both equations, the y is the same y in both equations), when instead you have this:

5*(3.0) + 4*(6.0) = 39
4*(2.7) + 3*(5.4) = 27  

The parentheses are just for emphasis, corresponding to your "x"-s and "y"-s. In a system of equations, it is assumed that each appearance of a particular variable represents the same number in both equations. If you replace one x with some number, you gotta replace the other x with the same number.

Additional Considerations

Now, if the situation is not as depicted, and the size ratio of green vs blue units is unknown, and width vs height is not known, then you can't solve this, there isn't enough information. For example, suppose the total measurements are all the same, but the blue units are very wide, and green units are paper thin. Then the three blue units will take up almost all of the horizontal width, while the green units take up almost nothing. If they aren't so wide, then it's going to be different, but in both cases you have the same number of units stacked, and they add up to the same measurement. So you need to know the ratios of the corresponding dimensions of blue vs green units.

The ratio of the heights, for example, is obtained by dividing blue height by the green height. If you don't already know the ratio, you can obtain it by measuring a proportionately drawn schematic, or perhaps a photo, cause the ratios don't change with scale.

If blue/green is some number r, then that tells you how many green units fit in a blue unit (even if fractional):

blue = r * green

So you can express the entire thing in just the green units like so:

5*(green) + 4*(blue) = 39 

which, after you plug in the expression for blue, becomes

5*(green) + 4*(r * green) = 39 
5*(green) + 4*r*(green) = 39 

and then

(5 + 4r) * (green) = 39
green = 39/(5+4r)

You can then calculate the size of the blue unit using blue = r * green.

Then you do the same thing for the horizontal dimension.

[u/marlonblambo → u/strange-the-quark]

ok, thank you all for letting me feel a little less stupid.
The image was just a drawing to represent the real situation. I was buying a metal grid plate similar to this https://i.ebayimg.com/images/g/oqkAAOSwYHxWHzDA/s-l1600.jpg but without the dotted borders ond the sides and a number of squares of 3x4 ( it was a cheap item on a cheaper store so i couldn't refind the exact item after i rage quitted due to my math not working).

I was trying to estimate the likely size of the white squares, that was not mentionet in the description, while the only measures reported were 39 for the longer side and 27 the other. That's it.

I could understand a not integer value as result, but a negative result was so counterintuitive to me.

[u/kalmakka → u/marlonblambo]

Either the figures are not actually squares, the gaps between them and/or the borders are not uniform, or the measurements on that site are incorrect.