Answer
$(-50+t, 170+t, 430-t, t)$, $50\leq t\leq430$
Work Step by Step
Step 1. Establish the system of equations, the input and output at each intersection should be equal:
$\begin{cases} 200+180=x+z \\ y+z=200+400 \\ x+70=w+20 \\w+200=y+30 \end{cases}$
Step 2. Simplify and rearrange these equations:
$\begin{cases} x+0y+z+0w=380 \\0x+ y+z+0w=600 \\ x+0y+0z-w=-50 \\0x+y+0z-w=170 \end{cases}$
Step 3. Establish the augmented matrix of the system and use the Gauss Eliminations method:
$\begin{vmatrix} 1 & 0 & 1 & 0 & 380\\ 0 & 1 & 1 & 0 & 600\\ 1 & 0 & 0 & -1 & -50\\ 0 & 1 & 0 & -1 & 170 \end{vmatrix} \begin{array}( \\ \\R_1-R_3\to R_3\\R_2-R_4\to R_4 \\ \end{array}$
Step 4. Do the operations given on the right side of the matrix.
$\begin{vmatrix} 1 & 0 & 1 & 0 & 380\\ 0 & 1 & 1 & 0 & 600\\ 0 & 0 & 1 & 1 & 430\\ 0 & 0 & 1 & 1 & 430 \end{vmatrix} \begin{array}( \\ \\ \\ \\ \end{array}$
Step 5. The last two rows are the same, let $w=t$, the third row gives $z+t=430$ and $z=430-t, (t\leq430)$, the second row gives $y+z=600$ and $y=170+t$, and the first row gives $x+z=380$ and $x=-50+t, (t\geq50)$
Step 6. The solutions are $(-50+t, 170+t, 430-t, t)$ with $50\leq t\leq430$
