To solve this problem, let's go step by step through each part:

a. Finding a formula expressing Celsius temperature $C$ as a linear function of Fahrenheit temperature $F$.

Since we know that the relationship between Celsius and Fahrenheit is linear, we can express this as a linear equation: $C = mF + b$ where:

• $C$ is the Celsius temperature,
• $F$ is the Fahrenheit temperature,
• $m$ is the slope,
• $b$ is the y-intercept.

We are given two points:

1. When water freezes: $F = 32$, $C = 0$,
2. When water boils: $F = 212$, $C = 100$.

Using these two points, we can find the slope $m$. The formula for the slope is: $m = \frac{C_2 - C_1}{F_2 - F_1}$ Substituting the points $(32, 0)$ and $(212, 100)$: $m = \frac{100 - 0}{212 - 32} = \frac{100}{180} = \frac{5}{9}$

Now, using the point-slope form of a linear equation, we'll use the point $(32, 0)$ to find the y-intercept $b$. The equation becomes: $C - 0 = \frac{5}{9}(F - 32)$ Simplifying: $C = \frac{5}{9}(F - 32)$ Thus, the linear formula for converting Fahrenheit to Celsius is: $C = \frac{5}{9}(F - 32)$

b. What is the slope of the function?

The slope $m$ is $\frac{5}{9}$.

Practical meaning:

The slope $\frac{5}{9}$ means that for every 1-degree increase in Fahrenheit, the Celsius temperature increases by $\frac{5}{9}$ degrees. This represents the rate at which Celsius temperature changes relative to Fahrenheit.

c. Solving the equation $F = 1.8C + 32$ for $C$.

We are given the equation: $F = 1.8C + 32$ To solve for $C$, subtract 32 from both sides: $F - 32 = 1.8C$ Now, divide both sides by 1.8: $C = \frac{F - 32}{1.8}$ Notice that $1.8 = \frac{9}{5}$, so we can rewrite the equation as: $C = \frac{5}{9}(F - 32)$ This is the same formula that we derived in part (a), confirming that both approaches yield the same result.

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Summary:

• Part a: The formula for Celsius in terms of Fahrenheit is $C = \frac{5}{9}(F - 32)$.
• Part b: The slope is $\frac{5}{9}$, which represents how much Celsius changes for every degree change in Fahrenheit.
• Part c: Solving $F = 1.8C + 32$ for $C$ gives the same formula: $C = \frac{5}{9}(F - 32)$.

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Do you want more details or have any questions? Here are 5 related questions to explore further:

1. How can you derive the Fahrenheit-to-Celsius conversion formula by graphing the points (32, 0) and (212, 100)?
2. What happens to the Celsius scale as the Fahrenheit temperature decreases below freezing?
3. Can you derive the reverse formula, converting Celsius to Fahrenheit, starting from the linear relationship found?
4. How does the slope $\frac{5}{9}$ relate to real-world temperature scales, such as weather changes between seasons?
5. Can you find an example where the temperature in Celsius is negative, and what would that imply in Fahrenheit?

Tip: When solving for one variable in terms of another, make sure to manipulate the equation step by step, keeping track of each transformation!