Certainly! Let's derive the formula for converting Fahrenheit (F) to Celsius (C) by finding the equation of the line that passes through the points (32, 0) and (212, 100). Here are the steps involved:
1. Identify the Points:
- Point 1: (32, 0)
- Point 2: (212, 100)
2. Calculate the Slope ([tex]\( m \)[/tex]):
- The slope formula is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Substituting the given points:
[tex]\[
m = \frac{100 - 0}{212 - 32}
\][/tex]
- Simplifying the calculation:
[tex]\[
m = \frac{100}{180} = \frac{5}{9}
\][/tex]
3. Find the Y-Intercept ([tex]\( b \)[/tex]):
- The equation of the line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
- We can use one of the points (32, 0) to find [tex]\( b \)[/tex]:
[tex]\[
0 = \left(\frac{5}{9}\right) \times 32 + b
\][/tex]
- Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 0 - \left(\frac{5}{9} \times 32\right) = -\frac{160}{9}
\][/tex]
4. Write the Equation:
- Now we have the slope [tex]\( m = \frac{5}{9} \)[/tex] and the y-intercept [tex]\( b = -\frac{160}{9} \)[/tex].
- Therefore, the equation in slope-intercept form is:
[tex]\[
C = \frac{5}{9}F - \frac{160}{9}
\][/tex]
So, the formula for converting Fahrenheit (F) to Celsius (C) by determining the line through the points (32, 0) and (212, 100) is:
[tex]\[
C = \frac{5}{9}F - \frac{160}{9}
\][/tex]
