jmc

$\text{Solution 1: Find the equation of the line.}$

A point-slope equation of the line is $y-0=-2(x-5),$ so the standard form of the equation of the line is $2x+y=10.$ When $x=0,$ we have $y=10,$ so the $y$-intercept is $\boxed{(0,10)}.$

$\text{Solution 2: Use the slope without finding the equation.}$

Because the slope of the line is $-2,$ the line goes down $2$ steps for each $1$ step right. However, the point we are given on the line, $(5,0),$ is already to the right of the $y$-axis, which is where the $y$-intercept is.Therefore, we instead think of the slope as going up $2$ steps for each $1$ step left. We must take $1$ step left $5$ times to get to the $y$-axis from point $(5,0),$ so to stay on this line, we must also take $2$ steps up $5$ times, for a total of $10$ steps. The point that is $10$ steps up and $5$ steps to the left of $(5,0)$ is $\boxed{(0,10)}.$