Find the slope of the line which contains the point (-4,7) and also contains the point at which the line 3x-2y=8 intersects the y-axis.?

Method 1 :

The two intersecting lines :
3x - 2y = 8 …… [1]
x = 0 …… [2]

Substitute [2] into [1] :
3(0) - 2y = 8
y = -4

The point of intersection = (0, -4)

The slope of the line containing (-4, 7) and (0, -4)
= [7 - (-4)] / (-4 - 0)
= -11/4

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Method 2 :

The line system passing through intersection of the line (3x - 2y = 8) and the y-axis :
(3x - 2y - 8) + kx = 0
(3 + k)x - 2y - 8 = 0 …… [1]

The required line passing through (-4, 7) :
Substitute x = -4 and y = 7 into [1] :
(3 + k)(-4) - 2(7) - 8 = 0
-12 - 4k - 14 - 8 = 0
4k = -34
k = -17/2

Substitute k = -34/12 into [1] :
[3 + (-17/2)]x - 2y + 8 = 0
[6 - 17]x - 4y + 8 = 0
-11x - 4y + 8 = 0
4y = -11x + 8
y = (-11/4)x + 2

The slope of the line containing (-4, 7) and (0, -4) = -11/4

3x -2y = 8
cross the y-axis when x = 0
-2y = 8
y = 8/-2 =-4
so find a line with points
(-4,7) and (0, -4)
y = mx +b
b = -4
m = (y2-y1) / (x2 -x1) = (7- (-4)) / ( (-4 - (0) ) = 11/-4

slope = m = -11/4
answer
slope (in decimal ) = -2.75
equation of the line
y = (-11/4)x - 4

3x - 2y = 8
cuts y axes at ( 0 , - 4)

m = [ - 4 - 7 ] / [ 4 ]
m = - 11 / 4
m = - 2 • 7 5

"the point at which the line 3x-2y = 8 intersects the y-axis"
That would be the y-intercept. Set x to 0 and solve for y.
3·0-2y = 8
y-intercept = -4

Now you want the slope of the line passing through (-4,7) and (0,-4).
rise = -4 - 7 = -11
run = 0 - (-4)= 4
slope = rise/run = -11/4 = -2.75