Answer
$a)$ $y=-4x$
$b)$ $4x+y=0$
$c)$
Work Step by Step
The line that passes through the origin and is parallel to the line containing $(2,4)$ and $(4,-4)$
$a)$
Since the line whose equation must be found is parallel to the line that passes through the points given, both lines have the same slope.
Use the points given to find the slope of the line parallel to the line whose equation must be found:
$m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}=\dfrac{-4-4}{4-2}=\dfrac{-8}{2}=-4$
The fact that the line whose equation must be found passes through the point $(0,0)$ is known. Its slope is also known since its the same slope that was just found.
Substitute these values into the point-slope form of the equation of a line formula, which is $y-y_{1}=m(x-x_{1})$, where $m$ is the slope of the line and $(x_{1},y_{1})$ is a point through which it passes, and simplify:
$y-0=-4(x-0)$
$y=-4x$
$b)$
To represent the equation of this line in general form, just take $-4x$ to the left side:
$4x+y=0$
$c)$
