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.