Chapter 3.4, Problem 51E

Competition A product is made by only two competing companies. Suppose Company $X$

Retains two-thirds of its customers and loses one-third to Company $Y$ each year, and Company $Y$ each year, and Company $Y$ retains three-quarters of its customers and loses one-quarter to Company $X$ each year. We can represent the number of customers each company had last year by

$\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]$

Where $x_0$ is the number Company $X$ had and $y_0$ is the number Company $Y$

had. Then the number that each will have this year can be represented by

$\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}\frac{2}{3}& \frac{1}{4}\\ \frac{1}{3}& \frac{3}{4}\end{array}\right]\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]$

If Company $X$ has $1900$ customers and Company $Y$ has $1700$ customers this year, how many customers did each have last year?