x*10000+(x+y)*1000+z=12; (x+y)*1000-w*39000+0.6=0; (y+w)*160000+w*39000=12; x+y=10^(-6)*120*(exp^(0.6/(2*25.5*10^(-3)))-1)

Given the system of equations
$$z + \left(10000 x + 1000 \left(x + y\right)\right) = 12$$
$$\left(- 39000 w + 1000 \left(x + y\right)\right) + \frac{3}{5} = 0$$
$$39000 w + 160000 \left(w + y\right) = 12$$
$$x + y = 0.00012 \left(-1 + e^{\frac{3}{5 \cdot 0.001 \frac{2 \cdot 51}{2}}}\right)$$

We give the system of equations to the canonical form
$$11000 x + 1000 y + z = 12$$
$$- 39000 w + 1000 x + 1000 y = - \frac{3}{5}$$
$$199000 w + 160000 y = 12$$
$$x + y = 15.4356426500989$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}0 & 11000 & 1000 & 1 & 12\\-39000 & 1000 & 1000 & 0 & - \frac{3}{5}\\199000 & 0 & 160000 & 0 & 12\\0 & 1 & 1 & 0 & \frac{108}{7}\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}0\\-39000\\199000\\0\end{matrix}\right]$$
let’s convert all the elements, except
3 -th element into zero.
- To do this, let’s take 3 -th line
$$\left[\begin{matrix}199000 & 0 & 160000 & 0 & 12\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}-39000 - \frac{\left(-39\right) 199000}{199} & 1000 - \frac{\left(-39\right) 0}{199} & 1000 - \frac{\left(-39\right) 160000}{199} & - \frac{\left(-39\right) 0}{199} & - \frac{3}{5} - \frac{\left(-39\right) 12}{199}\end{matrix}\right] = \left[\begin{matrix}0 & 1000 & \frac{6439000}{199} & 0 & \frac{1743}{995}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 11000 & 1000 & 1 & 12\\0 & 1000 & \frac{6439000}{199} & 0 & \frac{1743}{995}\\199000 & 0 & 160000 & 0 & 12\\0 & 1 & 1 & 0 & \frac{108}{7}\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}11000\\1000\\0\\1\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}0 & 1000 & \frac{6439000}{199} & 0 & \frac{1743}{995}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- 0 \cdot 11 & 11000 - 11 \cdot 1000 & 1000 - \frac{11 \cdot 6439000}{199} & 1 - 0 \cdot 11 & 12 - \frac{11 \cdot 1743}{995}\end{matrix}\right] = \left[\begin{matrix}0 & 0 & - \frac{70630000}{199} & 1 & - \frac{7233}{995}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 0 & - \frac{70630000}{199} & 1 & - \frac{7233}{995}\\0 & 1000 & \frac{6439000}{199} & 0 & \frac{1743}{995}\\199000 & 0 & 160000 & 0 & 12\\0 & 1 & 1 & 0 & \frac{108}{7}\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{0}{1000} & 1 - \frac{1000}{1000} & 1 - \frac{6439000}{199 \cdot 1000} & - \frac{0}{1000} & \frac{108}{7} - \frac{1743}{995 \cdot 1000}\end{matrix}\right] = \left[\begin{matrix}0 & 0 & - \frac{6240}{199} & 0 & \frac{107447799}{6965000}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 0 & - \frac{70630000}{199} & 1 & - \frac{7233}{995}\\0 & 1000 & \frac{6439000}{199} & 0 & \frac{1743}{995}\\199000 & 0 & 160000 & 0 & 12\\0 & 0 & - \frac{6240}{199} & 0 & \frac{107447799}{6965000}\end{matrix}\right]$$
In 3 -th column
$$\left[\begin{matrix}- \frac{70630000}{199}\\\frac{6439000}{199}\\160000\\- \frac{6240}{199}\end{matrix}\right]$$
let’s convert all the elements, except
4 -th element into zero.
- To do this, let’s take 4 -th line
$$\left[\begin{matrix}0 & 0 & - \frac{6240}{199} & 0 & \frac{107447799}{6965000}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{0 \cdot 882875}{78} & - \frac{0 \cdot 882875}{78} & - \frac{70630000}{199} - - \frac{70630000}{199} & 1 - \frac{0 \cdot 882875}{78} & - \frac{882875 \cdot 107447799}{78 \cdot 6965000} - \frac{7233}{995}\end{matrix}\right] = \left[\begin{matrix}0 & 0 & 0 & 1 & - \frac{181606939}{1040}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 0 & 0 & 1 & - \frac{181606939}{1040}\\0 & 1000 & \frac{6439000}{199} & 0 & \frac{1743}{995}\\199000 & 0 & 160000 & 0 & 12\\0 & 0 & - \frac{6240}{199} & 0 & \frac{107447799}{6965000}\end{matrix}\right]$$
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\left(-160975\right) 0}{156} & 1000 - \frac{\left(-160975\right) 0}{156} & \frac{6439000}{199} - - \frac{-6439000}{199} & - \frac{\left(-160975\right) 0}{156} & \frac{1743}{995} - \frac{\left(-160975\right) 107447799}{156 \cdot 6965000}\end{matrix}\right] = \left[\begin{matrix}0 & 1000 & 0 & 0 & \frac{1159015933}{72800}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 0 & 0 & 1 & - \frac{181606939}{1040}\\0 & 1000 & 0 & 0 & \frac{1159015933}{72800}\\199000 & 0 & 160000 & 0 & 12\\0 & 0 & - \frac{6240}{199} & 0 & \frac{107447799}{6965000}\end{matrix}\right]$$
From 3 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}199000 - \frac{\left(-199000\right) 0}{39} & - \frac{\left(-199000\right) 0}{39} & 160000 - - -160000 & - \frac{\left(-199000\right) 0}{39} & 12 - \frac{\left(-199000\right) 107447799}{39 \cdot 6965000}\end{matrix}\right] = \left[\begin{matrix}199000 & 0 & 0 & 0 & \frac{35821393}{455}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 0 & 0 & 1 & - \frac{181606939}{1040}\\0 & 1000 & 0 & 0 & \frac{1159015933}{72800}\\199000 & 0 & 0 & 0 & \frac{35821393}{455}\\0 & 0 & - \frac{6240}{199} & 0 & \frac{107447799}{6965000}\end{matrix}\right]$$

It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{4} + \frac{181606939}{1040} = 0$$
$$1000 x_{2} - \frac{1159015933}{72800} = 0$$
$$199000 x_{1} - \frac{35821393}{455} = 0$$
$$- \frac{6240 x_{3}}{199} - \frac{107447799}{6965000} = 0$$
We get the answer:
$$x_{4} = - \frac{181606939}{1040}$$
$$x_{2} = \frac{1159015933}{72800000}$$
$$x_{1} = \frac{180007}{455000}$$
$$x_{3} = - \frac{35815933}{72800000}$$