When temperature is zero degree Celsius, the Fahrenheit temperature is 32. When the Celsius temperature is 100, the corresponding Fahrenheit temperature is 212. Express the Fahrenheit temperature as a linear function of C, the Celsius temperature, F(c).

We are asked to express the Fahrenheit temperature as a function of Celsius temperature, so the independent variable, in this case, is x (Celsius temperature) and the dependent variable is y (Fahrenheit temperature).

When temperature is zero degree Celsius ([tex]\\x_1 = 0[/tex]), the Fahrenheit temperature is 32 ([tex]\\y_1 = 32[/tex]).

When the Celsius temperature is 100 ([tex]\\x_2 = 100[/tex]), the corresponding Fahrenheit temperature is 212 ([tex]\\y_2 = 212[/tex]).

Then, using [1], we have:

[tex]\\ y-32 = \frac{212 - 32}{100 - 0}(x-0)[/tex]

[tex]\\ y-32 = \frac{180}{100}(x)[/tex]

[tex]\\ y-32 = 1.8(x)[/tex].

It could be also be written as:

[tex]\\ y-32 = \frac{18}{10}(x)[/tex] = [tex]\\ y-32 = \frac{9}{5}(x)[/tex], as it commonly appears in books.

Then the Fahrenheit temperature express as a linear function of the Celsius temperature, F(c) is ( solving the equation for y ) :

[tex]\\ y = 1.8(x) + 32 [/tex] or [tex]\\ y = \frac{9}{5}(x) + 32 [/tex].

Or equivalently:

[tex]\\ F = 1.8(C) + 32[/tex] or [tex]\\ F = \frac{9}{5}(C) + 32 [/tex]

We can check this using the given values from the question:

For 0 Celsius degrees, the Fahrenheit temperature is:

[tex]\\ y = 1.8(0) + 32 [/tex] = 32 Fahrenheit degrees.

For 100 Celsius degrees, the Fahrenheit temperature is:

[tex]\\ y = 1.8(100) + 32 = 180 + 32 [/tex] = 212 Fahrenheit degrees.