write the standard form of equation

Question

through(-4,-5) perp to y=-4/7x-2

Answer 1

The problem gives the equation of a line in Slope-Intercept form: y=mx+b    where m=slope; b=y-intercept
 
The Standard Form of the equation of a line is:  Ax + By = C
Converting that, once and forever, into slope-intercept form, it is:
                     y = (-A/B)x + (C/B)
 
Now, the slope of a perpendicular line is the negative reciprocal of the slope of the line  (very important).
 
The line is:
      y = -4/7x - 2
In Standard Form:
     (4/7)x + y = -2
       4x + 7y = -14                  [multiply by 7 to get rid of fraction]
 
The line has slope m=(-A/B)=(-4/7).
The perpendicular line has slope=(7/4)=(-A/B), so the perpendicular line is:
          -7x + 4y = D              [we need point (-4,-5) to determine the value of D]
                                 (-7)(-4) + (4)(-5) = D
                                         28 - 20 = D
                                                  8 = D
         -7x + 4y = 8
The perpendicular line is:
          7x - 4y = -8             [make the x-coefficient positive by multiplying by (-1)]
 
Checking (very important):
  The slope of the line is (-A/B)=-7/4;   the slope of the perpendicular line is:  (-7)/(-4)=+7.4.
  Is the point (-4,-5) on the perpendicular line?
       7(-4) - 4(-5) = -8   ?
         -28  +  20  =  -8   ?
              -8   =  -8   ? yes