Different Scales, Equivalent Temperatures

The formulas for converting between degree Celsius and degree Fahrenheit are:

$\color{#3D99F6}{^\circ \text{F}} = \left( \frac{9 \color{#20A900}{^\circ \text{C}}}{5}\right) + 32$ $\color{#20A900}{^\circ \text{C}} =\frac{5}{9} \left(\color{#3D99F6}{^\circ\text{ F}} - 32\right)$

Setting $^\circ \text{F}=^\circ \text{C}$

$x {°C} = \left( \frac{9 x {^\circ \text{C}}}{5}\right) + 32$ $x{°C} - \left( \frac{9 x{^\circ \text{C}}}{5}\right) = 32$ $-\frac{4x{°C}}{5} = 32$ $x{°C} = -32 * \frac{5}{4}$ $x{°C} = -40$ $\implies x=\color{#D61F06}{\boxed{-40}}$

So the temperature when both the Celsius and Fahrenheit scales are the same is $-40$ degrees.