College Algebra, Chapter 2, 2.4, Section 2.4, Problem 70

The equation $\displaystyle F = \frac{9}{5} C + 32$ represents the relationship between Fahrenheit $(F)$ and Celsius $(C)$ temperature scales.

a.) Complete the table to compare the two scales at the given values.

b.) Find the temperature at which the scales agree.

a.)
$
\begin{array}{|c|c|}
\hline\\
C & F \\
\hline\\
-30^{\circ} & \underline{-22^{\circ}} \\
-20^{\circ} & \underline{-4^{\circ}} \\
-10^{\circ} & \underline{14^{\circ}} \\
0^{\circ} & \underline{32^{\circ}} \\
\underline{10^{\circ}} & 50^{\circ} \\
\underline{20^{\circ}} & 68^{\circ} \\
\underline{30^{\circ}} & 86^{\circ}\\
\hline
\end{array}
$


$
\begin{equation}
\begin{aligned}

F =& \frac{9}{5} C + 32
\\
\\
C =& \frac{5}{9} (^{\circ} F - 32)

\end{aligned}
\end{equation}
$


b.) If we set $F = T$ and $C = T$, then


$
\begin{equation}
\begin{aligned}

T =& \frac{9}{5}T + 32
\\
\\
T - \frac{9}{5} T =& 32
\\
\\
\frac{-4}{5} T =& 32
\\
\\
-4T =& 160
\\
\\
T =& -40^{\circ}

\end{aligned}
\end{equation}
$


It means that the temperature scales are equal at $-40^{\circ}$