3x+4y=7, 4y=-3x+7, y=-¾x+7/4, so the slope of the line is -¾. All lines parallel to the given line have the same slope.
y-q=m(x-p) is one way to express the equation of a line of slope m passing through point (p,q).
(a) When m=-¾, p=-⅔, q=⅞:
y-⅞=-¾(x+⅔)=-¾x-½.
Multiply through by 8: 8y-7=-6x-4, 6x+8y=3 is the equation of the parallel line.
This can also be written: y=-¾x+⅜ showing the slope and y-intercept.
(b) The slope of the perpendicular line is -1/m=4/3.
y-⅞=(4/3)(x+⅔)=4x/3+8/9. Multiply through by 72:
72y-63=96x+64, 96x-72y=-127 is the equation of the perpendicular line.
This can also be written: y=4x/3+127/72 showing the slope and y-intercept.