Show all work to write the equations of the lines, representing the following conditions, in the form y = mx + b, where m is the slope and b is the y-intercept: part a: passes through (−2, 2) and parallel to 4x − 3y − 7 = 0 part b: passes through (−2, 2) and perpendicular to 4x − 3y − 7 = 0

The general form given is:
y = mx + c where:
m is the slope 
c is the y-intercept

Part (a):
(i) getting the slope:
We are given that the line is parallel to:
4x - 3y - 7 = 0
This means that both lines have the same slope.
We will attempt to write the given equation in the standard form to get its slope as follows:
4x - 3y - 7 = 0
3y = 4x - 7
y = (4/3) x - 7/3
slope of the given line = slope of desired line = 4/3
(ii) getting the y-intercept:
We are given that (-2,2) belong to the line. To get the value of the c, we will substitute with this point in the equation and solve for c as follows:
y = mx + c
2 = (4/3)(-2) + c
c = 14/3
(iii) writing the equation of the line:
we calculated:
m = 4/3
c = 14/3
Therefore:
equation of line is:
y = (4/3) x + 14/3

Part (b):
(i) getting the slope:
We are given that the line is perpendicular to:
4x - 3y - 7 = 0
This means that slope of the desired line = -1/slope of given line.
We will attempt to write the given equation in the standard form to get its slope as follows:
4x - 3y - 7 = 0
3y = 4x - 7
y = (4/3) x - 7/3
slope of the given line = 4/3
slope of desired line = -3/4
(ii) getting the y-intercept:
We are given that (-2,2) belong to the line. To get the value of the c, we will substitute with this point in the equation and solve for c as follows:
y = mx + c
2 = (-3/4)(-2) + c
c = 1/2
(iii) writing the equation of the line:
we calculated:
m = -3/4
c = 1/2
Therefore:
equation of line is:
y = (-3/4) x + 1/2