Task $2.$

a$)$ In an industrial process, water flows through three tanks in succession, as illustrated in Figure $1.$

The tanks have unit cross$-$section and have heads of water $x,y$ and $z$ respectively $($cm$).$

The rate of inflow into the first tank is $u,$ in the second $v$ and in the third $w.$ The flowrate in the tube connecting tanks $1$ and $2$ is $8(x-y),$ the flowrate in the tube connecting tanks $2$ and $3$ is $6(y-z)$ and

Figure $1($not to scale$)$ the rate of outflow from tank $3$ is $9z.$

Show that the equations of the system in the steady flow situation are

$8x-8y=u$

$8x-14y+6z+v=0$

$6y-15z+w=0$

Take $u=100,v=30,w=20$ and find the values of $x,y$ and $z$ using Cramer's rule method.