The given line is 4x - 3y = 5.
Write the equation in standard form.
3y + 5 = 4x
3y = 4x - 5
[tex]y= \frac{4}{3}x- \frac{5}{3} [/tex]
This is a straight line with slope = 4/3, and with y-intercept = - 5/3.
A perpendicular line should have a slope of -3/4, because the product of the slopes of two perpendicular lines is -1.
Let the perpendicular line be
[tex]y=- \frac{3}{4}x+b [/tex]
Because the line passes through the point (3,1), therefore
[tex]- \frac{3}{4}(3)+b=1 \\\\ - \frac{9}{4}+b=1 \\\\ b=1+ \frac{9}{4}= \frac{13}{4} [/tex]
The equation of the perpendicular line is
[tex]y=- \frac{3}{4}x+ \frac{13}{4} [/tex]
The equation may also be written as
4y + 3x = 13.
A graph of the two lines is shown below.
Answer:
[tex]y=- \frac{3}{4}x+ \frac{13}{4} \,\, or \,\, 4y+3x=13 [/tex]
