#### Answer

$x-y=7$

#### Work Step by Step

Perpendicular lines have slopes that are negative reciprocals of each other.
The given line has a slope of $-1$.
This means that the slope of the line perpendicular to it is $1$.
Thus, the tentative equation of the line is $y=x+b$.
To find the value of $b$, substitute the x and y-coordinates of $(3, -4)$ into the tentative equation to have:
$y=x+b
\\-4 = 3+b
\\-4-3= b
\\-7=b$
Thus, the equation of the line parallel to the given line is $y=x-7$.
Convert this equation to $ax+by=c$ form to have:
$y=x-7
\\-x+y=-7
\\-1(-x+y)=-7(-1)
\\x-y=7$