call solving the linear system

$\frac{d{x}_1}{dt}=-\frac{1}{50}{x}_1$

$\frac{d{x}_2}{dt}=\frac{1}{50}{x}_1-\frac{2}{75}{x}_2$

$\frac{d{x}_3}{dt}=\frac{2}{75}{x}_2-\frac{1}{25}{x}_3$

ing elimination techniques. Recall that this linear system is a mathematical model for the number of pounds of salt $x_1(t),x_2(t),$ and $x_3(t)$ in the connected mixing tanks $A,B,$ an

$($a$)$ Use the eigenvalue method of this section to solve the system subject to $x_1(0)=20,x_2(0)=15,x_3(0)=10.$

$x_1(t)=$

$x_2(t)=$

$x_3(t)=$

$($b$)$ What are $\underset{t}{lim}{x}_1(t),\underset{t}{lim}{x}_2(t),$ and $\underset{t}{lim}{x}_3(t)?$

$\underset{t}{lim}{x}_1(t)=$

$\underset{t}{lim}{x}_2(t)=$

$\underset{t}{lim}{x}_3(t)=$