Name the line which have the following characteristics:

1) Passes through (2,3) & (0,-9)
2) Perpendiculat to the line 4x-2x=7 and through (1,-3)
3) Perpendicular to the segments with endpoints (2,1) & (-4,-1) and through the midpoint of the segment.

If possible show some quick work (not the full process)
thank you!!

1. slope = (3+9)/2-0) = 6

so (y-3)/(x-2)=6
cross-multiply, rearrange for a suitable version of the equation

2. slope of give line = 2, so slope of new line is -1/2
then (y+3)/(x-1) = -1/2
cross-multiply, rearrange for a suitable version of the equation

3. midpoint of segment = ((2-4)/2,(1-1)/2) = (-1,0)
slope of given segment = 1/3
so slope of new line = -3
so (y-0)/(x+1) = -3
cross-multiply, rearrange for a suitable version of the equation

To find the line with the given characteristics, we'll use the following methods:

1) To find the equation of the line passing through (2,3) and (0,-9), we can use the point-slope form of a linear equation. The formula is:
y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope.

Slope (m) = (change in y) / (change in x)
= (3 - (-9)) / (2 - 0)
= 12 / 2
= 6

Using the point-slope form with (x1, y1) = (2,3):
y - 3 = 6(x - 2)

2) The given line equation, 4x - 2x = 7, simplifies to 2x = 7. By solving for x, we find x = 7/2. Therefore, the slope of this line is 2.

Since the line we are looking for is perpendicular, the slope of the new line will be the negative reciprocal of 2, which is -1/2.

Using the point-slope form with (x1, y1) = (1,-3):
y - (-3) = -(1/2)(x - 1)

3) To find the slope of the line perpendicular to the segment with endpoints (2,1) and (-4,-1), we use the slope formula:
Slope = (1 - (-1)) / (2 - (-4))
= 2 / 6
= 1/3

Since the perpendicular line has the negative reciprocal slope of 1/3, the slope of the new line will be -3.

Since the line passes through the midpoint of the segment, we can find the midpoint using the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
= ((2 - 4) / 2, (1 - (-1)) / 2)
= (-1, 1/2)

Using the point-slope form with (x1, y1) = (-1, 1/2):
y - (1/2) = -3(x - (-1))

Therefore, the line equations with the given characteristics are:
1) y - 3 = 6(x - 2)
2) y + 3 = -(1/2)(x - 1)
3) y - (1/2) = -3(x - (-1))