The linear equation that converts Fahrenheit (F) to Celsius (C) is given by \[{\text{C = }}\dfrac{{\left( {{\text{5F - 160}}} \right)}}{{\text{9}}}\]
A) If the temperature is \[{86^\circ F}\], what is the temperature in Celsius?
B) If temperature is \[{35^\circ \text{C}}\], what is the temperature in Fahrenheit?
C) At what numerical value of temperature is the same in both the scales?

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Hint: Use the formula \[{\text{C = }}\dfrac{{\left( {{\text{5F - 160}}} \right)}}{{\text{9}}}\] and as given in various subparts if the temperature scale is in Fahrenheit than put that value in F or if temperature scale is in Celsius scale than put it in C and if both the temperature scale are equal than do F = C and proceed.

Complete step by step solution: Part – I
By using the given formula ,
\[{\text{C = }}\dfrac{{\left( {{\text{5F - 160}}} \right)}}{{\text{9}}}\]
As \[86^\circ {\text{ }}F\]is given and we have to find the temperature in degree Celsius
\[{\text{C = }}\dfrac{{\left( {{5 \times 86 - 160}} \right)}}{{\text{9}}}\]
\[{\text{ = }}\dfrac{{\left( {{\text{430 - 160}}} \right)}}{{\text{9}}}\]
\[{\text{ = }}\dfrac{{{\text{270}}}}{{\text{9}}}\]
\[{ = 30^\circ \text{C}}\]
By using the given formula,
\[{\text{C = }}\dfrac{{\left( {{\text{5F - 160}}} \right)}}{{\text{9}}}\]
C is \[35^\circ {\text{ }}C\], put C = 35 in the formula and simplify and calculate the value of F.
\[{\text{35 = }}\dfrac{{\left( {{\text{5F - 160}}} \right)}}{{\text{9}}}\]
\[{\text{5F = 35}}\left( {\text{9}} \right){\text{ + 160}}\]
  {{\text{F = 7}}\left( {\text{9}} \right){\text{ + 32}}} \\
  {{\text{F = 63 + 32}}}
\end{array} \\
  {\text{F = 95}^\circ \text{F}} \\
As now both the temperature scales are equal, proceed by doing C = F in the given formula.
Let \[{\text{F = C}}\] and put in the formula \[{\text{C = }}\dfrac{{\left( {{\text{5F - 160}}} \right)}}{{\text{9}}}\]
\[{\text{C = }}\dfrac{{\left( {{\text{5C - 160}}} \right)}}{{\text{9}}}\]
  {{\text{5C - 160 = 9C}}} \\
  {{\text{4C = - 160}}}
\end{array} \\
  {\text{C = - 40}^\circ \text{or F = - 40}^\circ } \\
At \[{- 40^\circ }\]temperature is the same in both the scales.

Note: The Celsius temperature range was originally defined by setting 0 as the temperature at which water freezes. Later it was redefined as the temperature at which ice melts. The other point at which Celsius was set was 100 Degrees Celsius, this was defined as the boiling point.
Whereas the Fahrenheit temperature range is based on setting the freezing point of water at 32 degrees, and the boiling point at 212 degrees.