Two lines are perpendicular when the multiplication of their slopes is equal to -1.
To find the slope of the line 2x - 3y = 4, we have to isolate y, as follows:
[tex]\begin{gathered} 2x-3y=4 \\ -3y=4-2x \\ y=\frac{4-2x}{-3} \\ y=\frac{4}{-3}+\frac{-2x}{-3} \\ y=-\frac{4}{3}+\frac{2}{3}x \end{gathered}[/tex]Its slope is 2/3, then the slope of the perpendicular line is:
[tex]\begin{gathered} m\cdot\frac{2}{3}=-1 \\ m=-1\cdot\frac{3}{2} \\ m=-\frac{3}{2} \end{gathered}[/tex]The slope-intercept form is:
y = mx + b
where m is the slope and b is the y-intercept.
Replacing with m = -3/2 and point (3,1):
1 = -3/2(3) + b
1 = -9/2 + b
1 + 9/2 = b
11/2 = b
The equation of the line is y = -3/2x + 11/2
