x + 2y + 3 = 0 (1)
2x - y + a/2 = 0 ⇒ 4x - 2y + a = 0 (2)
Add (1) and (2)
5x + (3 +a) = 0
x = -(3 + a) / 5
Sub this into (1) for x
-(3 + a) / 5 + 2y + 3 = 0
2y = -3 + (3 + a) / 5
y = (-12 + a) / 10
So......the intersection point is [x , y ] = [- (3 + a) /5 , (-12+ a) /10)]
The slope of the line through the given points is [ 9 - - 8 ] / [-17 - 9] = -17/26
And the equation of this line is
y = (-17/26) ( x -9) - 8
y = (-17/26) (x) + 153/26 -208/26
y = (-17/26)x - 55/26
Note that this line contains the intersection point of the given lines....so filling in the values for x and y
(-12 + a) / 10 = (-17/26) ( -(3 + a) / 5) - 55/26
Multiply through by 260
26 (-12 + a) = 2 ( -17) (-3-a) - 550
-312 + 26a = -34 (-3 - a) - 550
-312 + 26a = 102 + 34a - 550
-312 + 26a = 34a -448
136 = 8a
a =17
The intersection point is (-4, 1/2)
See the graph here : https://www.desmos.com/calculator/gsq2nbtdxq