Let's work through each of the problems step by step:

1. Equation of the line passing through (40, 9) and (10, -3)

We can use the formula for the equation of a line that passes through two points, which is the slope-intercept form:

$y - y_1 = m(x - x_1)$

where $m$ is the slope, and $(x_1, y_1)$ is one of the points. First, calculate the slope ($m$):

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 9}{10 - 40} = \frac{-12}{-30} = \frac{2}{5}$

Now use point-slope form with point $(10, -3)$:

$y + 3 = \frac{2}{5}(x - 10)$

Simplifying:

$y + 3 = \frac{2}{5}x - 4$

$y = \frac{2}{5}x - 7$

2. Equation with slope 12 and y-intercept of -8.7

The slope-intercept form of a line is:

$y = mx + b$

where $m$ is the slope and $b$ is the y-intercept. Given $m = 12$ and $b = -8.7$, the equation is:

$y = 12x - 8.7$

3. Equation with slope -5 passing through (2, 6)

We use point-slope form again:

$y - y_1 = m(x - x_1)$

Substitute $m = -5$ and $(x_1, y_1) = (2, 6)$:

$y - 6 = -5(x - 2)$

Simplifying:

$y - 6 = -5x + 10$

$y = -5x + 16$

4. Graph showing predicted outside temperature (time vs. temperature)

a. Predicted outside temperature at 12:00 PM
From the graph, we estimate the temperature at 12:00 PM. Based on the visual inspection, it looks like the temperature at 12:00 PM is around 20°F.

b. Time when the temperature is 0°F
Looking at the graph, the temperature reaches 0°F between 1:00 PM and 2:00 PM.

c. Equation to represent the line
To write an equation for the line, we first need two points from the graph. Suppose at 12:00 PM the temperature is 20°F, and at 3:00 PM it's -20°F. Using these two points, $(12, 20)$ and $(15, -20)$, we calculate the slope:

$m = \frac{-20 - 20}{15 - 12} = \frac{-40}{3}$

Now using point-slope form with point $(12, 20)$:

$y - 20 = \frac{-40}{3}(x - 12)$

Simplifying:

$y - 20 = \frac{-40}{3}x + 160$

$y = \frac{-40}{3}x + 180$

This is the equation representing the line for temperature prediction.

Do you want more details on any step? Here are some related questions to deepen understanding:

1. How do you determine the slope when given two points?
2. Why is the point-slope form useful for finding the equation of a line?
3. How does the slope affect the direction of the line?
4. Can a line have a slope of 0? What would its equation look like?
5. How can we use graphs to predict future values in real-world problems like temperature changes?

Tip: Remember, the slope of a line is a measure of how steep it is and indicates the rate of change between two variables.