Answer
$y=-14t+5815$
Work Step by Step
Let $y$ be the number of hospitals in the United States and $t$ be the number of years since $2000$. A linear equation for the number of hospitals $t$ years since $2000$ takes the form of
$$
y=mt+b,
$$where $m$ is the slope and $b$ is the $y$-intercept.
Since in $2001$ (i.e. $1$ year since $2000$) the number of hospitals is $5801$, this can be represented by the ordered pair $(t_1,y_1)=(1,5801)$. Since in $2004$ (i.e. $4$ years since $2000$) the number of hospitals is $5759$, this can be represented by the ordered pair $(t_2,y_2)=(4,5759)$.
The formula for finding the slope, $m$, of the line passing through two points, $(t_1,y_1)$ and $(t_2,y_2)$ is given by $m=\frac{y_1-y_2}{t_1-t_2}$. That is,
$$\begin{aligned}
m&=\frac{y_1-y_2}{t_1-t_2}
\\&=
\frac{5801-5759}{1-4}
\\&=
\frac{42}{-3}
\\&=
-14
.\end{aligned}
$$With $m=-14$, then the linear equation for the number of hospitals $t$ years since $2000$ takes the form of
$$
y=-14t+b
.$$Since the line passes through the point $(1,5801)$, substitute $t=1$ and $y=5801$ in the equation above to solve for $b$. That is,
$$\begin{aligned}
y&=-14t+b
\\
5801&=-14(1)+b
\\
5801&=-14+b
\\
5801+14&=b
\\
b&=5815
.\end{aligned}
$$With $b=5815$, then the linear equation for the number of hospitals $t$ years since $2000$ is
$$
y=-14t+5815
.$$
