###### Example4.6.8

A spa chain has been losing customers at a roughly constant rate since the year 2010. In 2013, it had $$2975$$ customers; in 2016, it had $$2585$$ customers. Management estimated that the company will go out of business once its customer base decreases to $$1800\text{.}$$ If this trend continues, when will the company close?

The given information tells us two points on the line: $$(2013,2975)$$ and $$(2016,2585)\text{.}$$ The slope formula (4.4.3) will give us the slope. After labeling those two points as $$(\overset{x_1}{2013},\overset{y_1}{2975})$$ and $$(\overset{x_2}{2016},\overset{y_2}{2585})\text{,}$$ we have:

\begin{align*} \text{slope}\amp=\frac{y_2-y_1}{x_2-x_1}\\ \amp=\frac{2585-2975}{2016-2013}\\ \amp=\frac{-390}{3}\\ \amp=-130 \end{align*}

And considering units, this means they are losing $$130$$ customers per year.

Let's note that we could try to make an equation for this line in slope-intercept form, but then we would need to calculate the $$y$$-intercept, which in context would correspond to the number of customers in year $$0\text{.}$$ We could do it, but we'd be working with numbers that have no real-world meaning in this context.

For point-slope form, since we calculated the slope, we know at least this much:

\begin{equation*} y=-130(x-x_0)+y_0\text{.} \end{equation*}

Now we can pick one of those two given points, say $$(2013,2975)\text{,}$$ and get the equation

\begin{equation*} y=-130(x-2013)+2975\text{.} \end{equation*}

Note that all three numbers in this equation have meaning in the context of the spa chain.

We're ready to answer the question about when the chain might go out of business. Substitute $$y$$ in the equation with $$1800$$ and solve for $$x\text{,}$$ and we will get the answer we seek.

\begin{align*} y\amp=-130(x-2013)+2975\\ \substitute{1800}\amp=-130(x-2013)+2975\\ 1800\subtractright{2975}\amp=-130(x-2013)\\ -1175\amp-130(x-2013)\\ \divideunder{-1175}{-130}\amp=x-2013\\ 9.038\amp\approx x-2013\\ 9.038\addright{2013}\amp\approx x\\ 2022\amp\approx x \end{align*}

And so we find that at this rate, the company is headed toward a collapse in 2022.

in-context