Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Digital systems, linear algebra, introduction to formal proof / 4 / b

Summing the flow conservation equations over all four junctions cancels every internal flow. The total entering flow is $300+500+250+50+100=1200$, and the known exiting flow is $700+200=900$. ThusThe original arrows point $A\to B$, $C\to A$, $B\to D$, and $C\to D$ for $f_2,f_3,f_4,f_5$, respectively. In particular, $f_1$ exits north from $A$.^[ref]

The individual flow conservation equations areChoose $t=f_5$. Substitution gives all solutions of these linear equations:The inequalities enforce nonnegative flows on all one-way streets. The last junction equation follows from the first three and total flow conservation, so it introduces no further restriction.

The maximum is $\boxed{f_5=150}$. It makes $f_2=f_3=0$. At $A$, all 300 vehicles arriving from the west turn north and leave along $f_1$; none arrives from $C$ or continues east to $B$.