Example $7\mathrm{.}3$ In the levelling network in Figure $7\mathrm{.}4,$ point $A$ is a benchmark and has an assumed height of $100\mathrm{.}00$ m $.$ Levelling has been undertaken along the lines as shown. The observed height differences were:

$\backslash$table$[[$Line$,\backslash$table$[[$Observed height$],[$difference$]],\backslash$table$[[$Approximate$],[$line length$]]],[Ax,12\mathrm{.}483$ m$,5$ km$],[AY,48\mathrm{.}351$ m$,10$ km$],[AZ,5\mathrm{.}492$ m$,7$ km$],[xY,35\mathrm{.}883$ m$,7$ km$],[xZ,-7\mathrm{.}093$ m$,12$ km$],[YZ,-42\mathrm{.}956$ m$,9$ km$]]$

The standard errors of the observed difference heights are believed to be:

${}_h=0\mathrm{.}017\sqrt[\phantom{2}]{?}Km,$ where $K$ is the line length in km

Find the best estimate of the heights of points $x,Y$ and $Z.$