To solve this problem, first we need to find the slope of the line between two points using the following formula:

Plug in (x1,y1)=(-2,-2) and (x2,y2)=(2,-4):

Next we need to remember point-slope formula:

Then we plug in m=-1/2 and (x1,y1)=(-2,-2) and solve:

To solve this problem, first we need to find the slope of the line between two points using the following formula:

Plug in (x2,y2)=(3,2) and (x1,y1)=(1,1):

Next we need to remember point-slope formula:

Then we plug in m=1/2 and (x2,y2)=(3,2) and solve:

To solve this problem, first we need to find the slope of the line between two points using the following formula:

Plug in (x2,y2)=(4,2) and (x1,y1)=(-2,1):

Next we need to remember point-slope formula:

Then we plug in m=1/6 and (x2,y2)=(4,2) and solve:

To solve this problem, first we need to find the slope of the line between two points using the following formula:

Plug in (x1,y1)=(3,-1) and (x2,y2)=(1,2):

Next we need to remember point-slope formula:

Then we plug in m=-3/2 and (x2,y2)=(1,2) and solve:

First, determine the number of cups of coffee with which Kim starts. The question says that Kim begins with three times more coffee than Ryan's initial four cups. Therefore, Kim begins with: 

 cups of coffee.

Next, determine the rate at which Kim drinks coffee. The question states that Kim drinks coffee half as fast as Ryan, who drinks the coffee at a rate of  cups per minute. Therefore, Kim drinks coffee at a rate of 

 cups per minute.

Now, realize that the initial amount of coffee represents the y-intercept (or c-intercept in this problem) as it is the y-value (c-value) when x=0 (m=0). Next, realize that the rate of Kim's coffee drinking is the slope of the equation.

In this instance, however, the coffee is disappearing as she drinks it, so the slope must be negative. Finally, put all of this information together using the slope-intercept form of a line to get: 

.

At first glance, it may seem that there is not enough information to determine the equation for either line. Notice, however, that the blue line passes through the origin, so we therefore know two points which lie on the blue line. These points are (0,0) and (2,4). Let's use these points to determine the slope of the blue line as such: .

Now, we may use this slope, to determine the slope of the perpendicular, red line. The slope of the red line is the negative reciprocal of the slope of the blue line because they are normal to one another. Therefore, the slope of the red line is 

.

Additionally, we are given a point through which the red line passes. This point is (0,3). We may immediately identify that this point represents the y-intercept, and so the y-intercept of the red line is . 

Let's now use the information about the red line to write the equation of that line in slope-intercept form as: 

.

When finding the equation of a line given two points, we must first find the slope of the line.  To find the slope of a line in slope-intercept form

where m is the slope, we calculate

where  and  are the points.  Given the points, we can substitue. We get

Now that we know the slope, we can substitute the slope into the slope-intercept form

All we need to do now if find the value of b.  To do that, we will substitute one of the points into the equation.  Let's use (4, 5).  So,

So if we substitute the value of b into the slope-intercept form with the slope included, we get

which can also be written as 

If we chose to use the slope formula to determine a possible slope, we have:

Our slope becomes undefined.  The x-values will not change, and we are not able to write this in the slope-intercept form.

This line is a vertical line, and will be represented by  because the x-values will not change in a vertical line.

The answer is: 

To determine the equation for the line given only a point on the line and its slope, we can use point-slope form, which is given by the following:

, where  is the slope of the line and  is the point on the line.

Using the formula, we get

, which simplified becomes .

Write the slope-intercept form for linear equations.

The  is the slope of the equation, and  is the y-intercept.

Substitute the values into the equation.

The equation of the line is: