Customize cases environment

Question

I usually use the cases environment in the amsmath package - it is very good:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[\begin{cases}
    x+y=2 \\
    2x+3y=4
\end{cases}\]
\end{document}

Pic 1

---

Question 1

However, in my country, the symbol doesn't really look like this. It is like this one instead - a slight change in the bracket:

Is it possible to redefine the cases environment so that it is like this:

Pic 2

Pic 3

Now honestly I have no idea of how to do this.

---

Question 2

Beside the { we also have the [. Now, I am using the [ like in cases environment using gathered, also in the amsmath package:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[
\left[\begin{gathered}
    x=3\hfill\\
    x=4\hfill
\end{gathered}\right.
\]
\end{document}

Pic 4

However, like Question 1 above, I want to get this kind of square bracket:

Pic 5

How to do so? I'm sorry I don't have the slightest idea :(

Question 2.1

Furthermore, in my country, people prefer this

Pic 6

Similarly to the above questions, I found many difficulties and failed.

Any help in any question will be highly appreciated!

---

Important Note

Pictures 2, 3, 5 and 6 are made with Windows Paint. As the spacing between lines in Euclid font is large, the spacing between two lines in the pictures are larger than I want. However the spacing I want is just normal, not that large. The default spacing in normal cases is perfect in this case.

Answer 1

It's relatively simple using \tikzmatrix.

I created two new environments (mycurly and mybrack) with an optional argument to indicate how many are the equations, in case they are more than two.

The code

\usepackage{expl3}
\ExplSyntaxOn
\int_new:N \g__prg_map_int 
\ExplSyntaxOff

is needed only if you have an error like this:

! Undefined control sequence.
<argument> \g__prg_map_int

(Thank you to Joseph Wright for solving the issue.)

\documentclass[12pt,fleqn]{article}
\usepackage{amsmath}
\usepackage{environ}
\usepackage{tikz}
\usetikzlibrary{matrix}
% the following code is needed if you got the error 
% ! Undefined control sequence.
% <argument> \g__prg_map_int 
% see here 
%https://tex.stackexchange.com/questions/463282/undefined-control-sequence-g-prg-map-int
% otherwise, leave it out
\usepackage{expl3}
\ExplSyntaxOn
\int_new:N \g__prg_map_int 
\ExplSyntaxOff
% end of temporary code
\usetikzlibrary{decorations.pathreplacing, calligraphy, calc}
\tikzset{
    mymatr/.style={
        matrix of math nodes, 
        column sep = 10pt,
        nodes={
            anchor=text, 
            inner xsep=3pt
        }, 
        inner xsep=7pt,
        ampersand replacement={\&},
    }
}
\NewEnviron{mybrack}[1][2]{%
    \begin{tikzpicture}[baseline=(mybase)]
    \matrix[mymatr](A){
        \BODY\\
    };
    \coordinate (mybase) at ($(A-1-1.text)!.5!(A-#1-1.text)$);
    \draw[thick]([xshift=-6.4pt]A-1-1.west) -- +(4.4pt,0);
    \foreach[evaluate=\mystep as \prev using int(\mystep-1)] \mystep in 
    {2,...,#1}{
        \draw[thick]([xshift=-6pt]A-\prev-1.west) -- 
        ([xshift=-6pt]A-\mystep-1.west) -- + (4pt,0);
    }
    \end{tikzpicture}
}

\NewEnviron{mycurly}[1][2]{%
    \begin{tikzpicture}[baseline=(mybase)]
    \matrix[mymatr](A){
        \BODY\\
    };
    \coordinate (mybase) at ($(A-1-1.text)!.5! (A-#1-1.text)$); 
    \draw[line width=.9pt,decorate,decoration={calligraphic 
    brace,amplitude=4pt, mirror}](A-1-1.west) -- (A-#1-1.west);
    \end{tikzpicture}
}


\begin{document}
    Environment which puts curly braces:
    \[
    \begin{mycurly}
    \frac{3}{4}x+y=2 \\
    e^{2x}+3y=4
    \end{mycurly}
    \]
    if you have more than 2 equation, you have to indicate their number in the 
    optional argument:
    \[
    \begin{mycurly}[3]
    \frac{3}{4}x+y=2 \\
    e^{2x}+3y=4\\
    \sqrt{x+3}+5z=0
    \end{mycurly}
    \]
    \[
    \begin{mycurly}[4]
    x+y=2 \\
    2x+3y=4 \\
    3x+5z=0 \\
    \dfrac{7x}{x+9}+y+z=2
    \end{mycurly}
    \]
    Environment which puts square brackets:
    \[
    \begin{mybrack}
    \frac{3}{4}x+y=2 \\
    e^{2x}+3y=4
    \end{mybrack}
    \]
    if you have more than 2 equation, you have to indicate their number in the 
    optional argument:
    \[
    \begin{mybrack}[3]
    \frac{3}{4}x+y=2 \\
    e^{2x}+3y=4\\
    \sqrt{x+3}+5z=0
    \end{mybrack}
    \]
    \[
    \begin{mybrack}[4]
    x+y=2 \\
    2x+3y=4 \\
    3x+5z=0 \\
    \dfrac{7x}{x+9}+y+z=2
    \end{mybrack}
    \]
    Please note you have to use \verb|\&| instead of \verb|&| here:
    \[
    f(x) = 
    \begin{mycurly}[4]
    \dfrac{1}{3}x+y=2 \& \text{if }x<0 \\
    2x+3y=4 \& \text{if }0\leq x<1\\
    3x+5z=0 \& \text{if }1\leq x<2\\
    \dfrac{7x}{x+9}+y+z=2 \& \text{if } x>2
    \end{mycurly}
    \]
    \[
    f(x) = 
    \begin{mycurly}[3]
    2x+3y=4 \& \text{if } x<1\\
    3x+5z=0 \& \text{if }1\leq x<2\\
    \dfrac{7x}{x+9}+y+z=2 \& \text{if } x>2
    \end{mycurly}
    \]
    It works also with the bracket type (even if I don't know if it is used in 
    your country):
    \[
    f(x) = 
    \begin{mybrack}[4]
    \dfrac{1}{3}x+y=2 \& \text{if }x<0 \\
    2x+3y=4 \& \text{if }0\leq x<1\\
    3x+5z=0 \& \text{if }1\leq x<2\\
    \dfrac{7x}{x+9}+y+z=2 \& \text{if } x>2
    \end{mybrack}
    \]
    \[
    f(x) = 
    \begin{mybrack}[3]
    2x+3y=4 \& \text{if } x<1\\
    3x+5z=0 \& \text{if }1\leq x<2\\
    \dfrac{7x}{x+9}+y+z=2 \& \text{if } x>2
    \end{mybrack}
    \]
\end{document}