###### Example4.6.3

Since 1990, the population of the United States has been growing by about $$2.865$$ million per year. Also, back in 1990, the population was $$253$$ million. Since the rate of growth has been roughly constant, a linear model is appropriate. But let's look for a way to write the equation other than slope-intercept form. Here are some things we know:

1. The slope equation is $$m=\frac{y_2-y_1}{x_2-x_1}\text{.}$$

2. The slope is $$m=2.865$$ (million per year).

3. One point on the line is $$(1990,253)\text{,}$$ since in 1990, the population was $$253$$ million.

If we use the generic $$(x,y)$$ to represent a point somewhere on this line, then the rate of change between $$(1990,253)$$ and $$(x,y)$$ has to be $$2.865\text{.}$$ So

\begin{equation*} \frac{y-253}{x-1990}=2.865\text{.} \end{equation*}

There is good reason 1  to want to isolate $$y$$ in this equation:

\begin{align*} \frac{y-253}{x-1990}\amp=2.865\\ y-253\amp=2.865\multiplyright{(x-1990)}\amp\amp\text{(could distribute, but not going to)}\\ y\amp=2.865(x-1990)\addright{253} \end{align*}

This is a good place to stop. We have isolated $$y\text{,}$$ and three meaningful numbers appear in the population: the rate of growth, a certain year, and the population in that year. This is a specific example of point-slope form.

in-context