Question

For what temperature value are the celsius value and the Fahrenheit value the same?

Answer 1

For this, we will need to find an expression for the celsius scale and set it equal to the fahrenheit scale.

Let `T_"F"` be the temperature on the fahrenheit scale, and `T_"C"` be the temperature on the celsius scale.

Recall that `\mathbf(T_F = 9/5T_C + 32)`.

The trick is to start by assuming that `T_F = T_C` (that is the premise of the problem) and finding `T_C` such that that is the case.

If `T_F = T_C`, then:

`T_C = 9/5T_C + 32`

`5/5T_C = 9/5T_C + 32`

`-32 = 4/5T_C`

`color(blue)(T_C) = -32*5/4`

`= color(blue)(-40^@ "C")`

Now, let's test this claim by converting back to fahrenheit using the first equation we looked at.

`color(blue)(T_F) = 9/5*(-40^@ "C") + 32`

`= -72 + 32`

`= color(blue)(-40^@ "F")` `color(green)(sqrt"")`

Given that we were solving a linear equation, there exists one and only one solution. Thus, we have found all solutions.