The formula to convert [tex]\ºC[/tex] to [tex]\ºF[/tex] is:
[tex]\ºF=\frac{9}{5}\ºC+32[/tex] (1)
and to convert [tex]\ºF[/tex] to [tex]\ºC[/tex] is:
[tex]\ºC=(\ºF-32)\frac{9}{5}[/tex] (2)
Now, if we want to know at what temperature does the numerical value in [tex]\ºF[/tex] match the numerical value in [tex]\ºC[/tex], we have to make [tex]\ºC=\ºF[/tex] and solve any of both equations shown above.
Using (1):
[tex]\ºC=\frac{9}{5}\ºC+32[/tex]
[tex]\ºC-\frac{9}{5}\ºC=32[/tex]
[tex]-\frac{4}{5}\ºC=32[/tex]
[tex]\ºC=-40[/tex]
Using (2):
[tex]\ºC=(\ºC-32)\frac{9}{5}[/tex]
[tex]\ºC-\frac{5}{9}\ºC=-\frac{160}{9}[/tex]
[tex]\ºC=-40[/tex]
