You notice that temperatures in Canada are reported in degrees Celsius. You know there is a linear relationship between degrees Fahrenheit and degrees Celsius, but you forget the formula. From science class, you remember the freezing point of water is 0°C or 32°F, and its boiling point is 100°C or 212°F.

a. Write an equation that represents degrees Fahrenheit in terms of degrees Celsius. Let $x$ represent the temperature in degrees Celsius and let $y$ represent the temperature in degrees Fahrenheit.

Equation: 
c. Rewrite your equation in part (a) to represent degrees Celsius in terms of degrees Fahrenheit. Use simplest fraction form when necessary. 
Equation: 

### Part a: Equation representing degrees Fahrenheit in terms of degrees Celsius
We know that there is a linear relationship between degrees Fahrenheit and degrees Celsius, and we are given two key points:

- Freezing point: $ (0, 32) $ where 0°C = 32°F
- Boiling point: $ (100, 212) $ where 100°C = 212°F

Using these points, we can find the slope $ m $ of the line that represents the relationship between Fahrenheit $ y $ and Celsius $ x $:

\[
m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8
\]

Now, using the point-slope form of a line, $ y - y_1 = m(x - x_1) $, we substitute $ m = 1.8 $, and one of the points, say $ (0, 32) $:

\[
y - 32 = 1.8(x - 0)
\]

Simplifying:

\[
y = 1.8x + 32
\]

Thus, the equation representing degrees Fahrenheit in terms of degrees Celsius is:

\[
y = 1.8x + 32
\]

### Part c: Equation representing degrees Celsius in terms of degrees Fahrenheit
To solve for $ x $ (degrees Celsius) in terms of $ y $ (degrees Fahrenheit), we start with the equation:

\[
y = 1.8x + 32
\]

Now, solve for $ x $:

1. Subtract 32 from both sides:
   \[
   y - 32 = 1.8x
   \]

2. Divide both sides by 1.8:
   \[
   x = \frac{y - 32}{1.8}
   \]

This can also be written in fraction form as:

\[
x = \frac{5}{9}(y - 32)
\]

Thus, the equation representing degrees Celsius in terms of degrees Fahrenheit is:

\[
x = \frac{5}{9}(y - 32)
\]

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Let me know if you want more details or have any questions! 

### Related Questions:
1. What is the temperature in Fahrenheit when the temperature in Celsius is 25°C?
2. If the temperature is 77°F, what is the temperature in Celsius?
3. How would you describe the relationship between Fahrenheit and Celsius in terms of proportionality?
4. What is the significance of the slope 1.8 in the context of temperature conversion?
5. How do the intercept values change if we shift the freezing point?

**Tip:** Always remember that the conversion formulas between Celsius and Fahrenheit are linear, meaning any increase or decrease in temperature will follow a consistent ratio.

You notice that temperatures in Canada are reported in degrees Celsius. You know there is a linear relationship between degrees Fahrenheit and degrees Celsius, but you forget the formula. From science class, you remember the freezing point of water is 0°C or 32°F, and its boiling point is 100°C or 212°F.

Write an equation that represents degrees Fahrenheit in terms of degrees Celsius. Let x represent the temperature in degrees Celsius and let y represent the temperature in degrees Fahrenheit.

Rewrite your equation to represent degrees Celsius in terms of degrees Fahrenheit. Use simplest fraction form when necessary.

Learn how to convert temperatures between Fahrenheit and Celsius using linear relationships. This problem involves deriving the formula y = 1.8x + 32 and solving for Celsius in terms of Fahrenheit. Suitable for students in Grades 8-10.

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