When the Celsius temperature is [tex]15^{\circ}[/tex], the corresponding Fahrenheit temperature is [tex]59^{\circ}[/tex]. When the Celsius temperature is [tex]120^{\circ}[/tex], the corresponding Fahrenheit temperature is [tex]248^{\circ}[/tex]. Let [tex]C[/tex] represent the Celsius temperature and [tex]F[/tex] the Fahrenheit temperature.

a. Express [tex]F[/tex] as an exact linear function of [tex]C[/tex].
[tex] F = \square [/tex]
(Simplify your answer. Do not include the degree symbol in your answer.)

b. Solve the equation in part (a) for [tex]C[/tex], thus expressing [tex]C[/tex] as a function of [tex]F[/tex].

c. For what temperature is [tex]F = C[/tex]?

Sure, let's go through the solution step-by-step for each part of the question.

### Part a: Express [tex]\( F \)[/tex] as an exact linear function of [tex]\( C \)[/tex].

We know two points with their corresponding Celsius ([tex]\( C \)[/tex]) and Fahrenheit ([tex]\( F \)[/tex]) values:
- When [tex]\( C = 15 \)[/tex], [tex]\( F = 59 \)[/tex]
- When [tex]\( C = 120 \)[/tex], [tex]\( F = 248 \)[/tex]

We can use these points to find the linear relationship [tex]\( F = mC + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.

Step 1: Calculate the slope ([tex]\( m \)[/tex])

The slope [tex]\( m \)[/tex] is calculated by the formula:
[tex]\[
m = \frac{F_2 - F_1}{C_2 - C_1}
\][/tex]

Substitute the given values:
[tex]\[
m = \frac{248 - 59}{120 - 15} = \frac{189}{105} = 1.8
\][/tex]

Step 2: Find the y-intercept ([tex]\( b \)[/tex])

We use one of the points to find [tex]\( b \)[/tex]. Let's use [tex]\( C = 15 \)[/tex] and [tex]\( F = 59 \)[/tex]:
[tex]\[
59 = 1.8(15) + b
\][/tex]
[tex]\[
59 = 27 + b
\][/tex]
[tex]\[
b = 59 - 27 = 32
\][/tex]

Thus, the linear function of [tex]\( F \)[/tex] in terms of [tex]\( C \)[/tex] is:
[tex]\[
F = 1.8C + 32
\][/tex]

### Part b: Solve the equation in part a for [tex]\( C \)[/tex], thus expressing [tex]\( C \)[/tex] as a function of [tex]\( F \)[/tex].

We have the equation:
[tex]\[
F = 1.8C + 32
\][/tex]

To express [tex]\( C \)[/tex] as a function of [tex]\( F \)[/tex]:

Step 1: Isolate [tex]\( C \)[/tex]

Rearrange the formula to solve for [tex]\( C \)[/tex]:
[tex]\[
F - 32 = 1.8C
\][/tex]
[tex]\[
C = \frac{F - 32}{1.8}
\][/tex]

So, [tex]\( C \)[/tex] in terms of [tex]\( F \)[/tex] is:
[tex]\[
C = \frac{F - 32}{1.8}
\][/tex]

### Part c: For what temperature is [tex]\( F = C \)[/tex]?

This means we need to find the temperature where the Fahrenheit and Celsius values are the same. Set [tex]\( F = C \)[/tex] in the formula:
[tex]\[
F = 1.8C + 32
\][/tex]

Since [tex]\( F = C \)[/tex], replace [tex]\( F \)[/tex] with [tex]\( C \)[/tex]:
[tex]\[
C = 1.8C + 32
\][/tex]

Rearrange to solve for [tex]\( C \)[/tex]:
[tex]\[
C - 1.8C = 32
\][/tex]
[tex]\[
-0.8C = 32
\][/tex]
[tex]\[
C = \frac{32}{-0.8} = -40
\][/tex]

So, the temperature where [tex]\( F = C \)[/tex] is:
[tex]\[
C = -40
\][/tex]

To summarize:

a. The linear function for [tex]\( F \)[/tex] in terms of [tex]\( C \)[/tex] is:
[tex]\[
F = 1.8C + 32
\][/tex]

b. The function for [tex]\( C \)[/tex] in terms of [tex]\( F \)[/tex] is:
[tex]\[
C = \frac{F - 32}{1.8}
\][/tex]

c. The temperature where [tex]\( F = C \)[/tex] is:
[tex]\[
C = -40
\][/tex]