Linear Relationship

A verbal description of the relationship

The dependent variable (Y) starts at some particular value and then increases or decreases by the same amount for each change in the independent value (X).

The equation and the meaning of x, y, and other parameters

Y = mX + b

m is the slope (change in in Y divided by the change in X)

b is the Y intercept, the value of Y when X = 0

One or more examples

A bucket already has 4 liters in it and you fill it from a hose that is flowing at 0.5 liters per minute.

Y is the volume of water in the bucket.

X is time in minutes

m is the rate of flow (0.5 liters/minute)

You can check the dimensions on this problem as follows:

Y (liters) = m (liters/min) * X (minutes) + b (liters)

all the terms in the equation simplify to liters

A graph

Algebraic rules that apply to the use of this equation

The following manipulations might be used

simple rearrangement such as Y - b = m*X, or (Y-b)/m = X

the logistic equation for growth rate as a function of population is usually given as

growth_rate = maximum_growth_rate * (K - N)/K

this represents the variable growth rate as a function of a constant maximum rate times a term (K-N)/K that is one when population =0 and goes to 0 when the population = the carrying capacity.

This equation can be rearranged to be in the standard linear form in the following steps:

start	growth_rate = maximum_growth_rate * (K - N)/K
expand the K-N term	growth_rate = maximum_growth_rate * K/K - maximum_growth_rate* N/K
rearrange	growth_rate = - maximum_growth_rate* N/K + maximum_growth_rate * K/K
simplify K/K =1	growth_rate = - maximum_growth_rate* N/K + maximum_growth_rate
isolate slope term	growth_rate = (- maximum_growth_rate/K) * N + maximum_growth_rate
end	Y = growth_rate X = N m = (- maximum_growth_rate/K) b = maximum_growth_rate

Characteristic values

at X = 0, Y = b

at Y = 0, X = -b/m

X and Y can have any values

with increasing values of X, Y will continue to increase or decrease with the same slope