Answer
See the detailed answer below,
Work Step by Step
a) We need to find the boiling point of water on the Z scale, and we know that the temperature scale is linear. So, we can represent it by the straight-line equation
$$y=ax+b$$
where $y$ here is the Celesius degree, $x$ here is the $Z$ degree.
Hence,
$$T_{\rm C}=aT_{\rm Z}+b\tag 1$$
Now we know that the boiling point of nitrogen is $0^\circ\rm Z$, and in Celsius, it is $−196^\circ\rm C$, So plug these into (1).
$$-196=a(0)+b$$
Hence,
$$b=-196 $$
Plugging into (1);
$$T_{\rm C}=aT_{\rm Z}-196 \tag 2$$
We also know that the melting point of iron is $1000^\circ\rm Z$, and in Celsius, it is $1538^\circ\rm C$.
Plugging into (2);
$$1538 =a(1000)-196 $$
Hence,
$$a=\dfrac{1538+196}{1000}=1.734$$
Therefore,
$$\boxed{T_{\rm C}=1.734\;T_{\rm Z}-196}$$
Now we can find the boiling point of water in $^\circ \rm Z$, where we know it is $ 100 ^\circ\rm C$.
$$100=1.734\;T_{\rm Z}-196$$
Thus,
$$T_{\rm Z}=\dfrac{100+196}{1.734}\approx \color{red}{\bf 171}^\circ\rm Z$$
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b) Plugging the $500^\circ\rm Z$ into the boxed formula above, to find it in Celsius.
$$ T_{\rm C}=1.734(500)-196=\color{red}{\bf 671}^\circ\rm C $$
Now we know that
$$T_{\rm K}=T_{\rm C}+273$$
Thus,
$$T_{\rm K}= 671+273=\color{red}{\bf 944} \rm\; K $$
