The problem asks: At what temperature is the difference between Fahrenheit and Celsius equal to 50 degrees? We are looking for a temperature $T$ such that $|F - C| = 50$, where $F$ is the Fahrenheit temperature and $C$ is the Celsius temperature.

1. Problem Description

The problem asks: At what temperature is the difference between Fahrenheit and Celsius equal to 50 degrees? We are looking for a temperature $T$ such that $|F - C| = 50$, where $F$ is the Fahrenheit temperature and $C$ is the Celsius temperature.

2. Solution Steps

First, we need to know the relationship between Fahrenheit and Celsius.

The formula to convert Celsius to Fahrenheit is:

$F = \frac{9}{5}C + 32$

We are given that $|F - C| = 50$. This means either $F - C = 50$ or $C - F = 50$.

Case 1: $F - C = 50$.

Substitute $F = \frac{9}{5}C + 32$ into the equation:

$(\frac{9}{5}C + 32) - C = 50$

$\frac{9}{5}C - C = 50 - 32$

$\frac{4}{5}C = 18$

$C = \frac{5}{4} \cdot 18 = \frac{90}{4} = \frac{45}{2} = 22.5$

Now, find $F$:

$F = C + 50 = 22.5 + 50 = 72.5$

Also, we can verify with $F = \frac{9}{5}C + 32$:

$F = \frac{9}{5}(22.5) + 32 = 9(4.5) + 32 = 40.5 + 32 = 72.5$

Case 2: $C - F = 50$.

Substitute $F = \frac{9}{5}C + 32$ into the equation:

$C - (\frac{9}{5}C + 32) = 50$

$C - \frac{9}{5}C - 32 = 50$

$-\frac{4}{5}C = 82$

$C = -\frac{5}{4} \cdot 82 = -\frac{410}{4} = -\frac{205}{2} = -102.5$

Now, find $F$:

$F = C - 50 = -102.5 - 50 = -152.5$

Also, we can verify with $F = \frac{9}{5}C + 32$:

$F = \frac{9}{5}(-102.5) + 32 = 9(-20.5) + 32 = -184.5 + 32 = -152.5$

Therefore, we have two solutions where the difference between Fahrenheit and Celsius is 50 degrees: $C = 22.5$ and $F = 72.5$, or $C = -102.5$ and $F = -152.5$. The question does not specify whether we want Celsius or Fahrenheit.

3. Final Answer

$22.5$ degrees Celsius (which is $72.5$ degrees Fahrenheit), or $-102.5$ degrees Celsius (which is $-152.5$ degrees Fahrenheit).