To find the linear equation that expresses the relationship between degrees Celsius ($C$) and degrees Fahrenheit ($F$), we use the known points of water freezing and boiling:

1. Water freezes at $0^\circ C$ and $32^\circ F$.
2. Water boils at $100^\circ C$ and $212^\circ F$.

The equation of a line is typically in the form:

$F = mC + b$

Where:

• $F$ is the temperature in Fahrenheit.
• $C$ is the temperature in Celsius.
• $m$ is the slope.
• $b$ is the y-intercept (the value of $F$ when $C = 0$).

Step 1: Find the slope $m$

Using the two points $(0, 32)$ and $(100, 212)$, we can find the slope $m$:

$m = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8$

Step 2: Find the y-intercept $b$

Since we know that when $C = 0$, $F = 32$, the y-intercept $b = 32$. So the equation becomes:

$F = 1.8C + 32$

Step 3: Solve for $C$

To convert Fahrenheit to Celsius, we can rearrange the equation to solve for $C$:

$C = \frac{F - 32}{1.8}$

Step 4: Convert 72°F to Celsius

Using the equation $C = \frac{F - 32}{1.8}$, we substitute $F = 72$:

$C = \frac{72 - 32}{1.8} = \frac{40}{1.8} \approx 22.22^\circ C$

Final Answer:

The temperature $72^\circ F$ is approximately $22.22^\circ C$.

Would you like further details or have any questions?

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Here are 5 related questions:

1. How would the formula change if we were converting from Kelvin to Celsius?
2. What is the temperature in Fahrenheit for $25^\circ C$?
3. How do you derive the slope formula for a linear equation from two points?
4. What is the interpretation of the slope $1.8$ in the context of this temperature conversion?
5. Can we use this linear equation for extreme temperatures, such as the temperature of the sun?

Tip: For quick temperature conversions between Celsius and Fahrenheit, remember the values 32°F (freezing) and 212°F (boiling) for reference points.