The intersection of the planes 8x + y - 12z = 35 and 6x + 7y - 9z = 70 is a line. Find the point where this line intersects the xy-plane.

Let z = t

So we have

8x + y  =  35 + 12t     ⇒    -56x - 7y  =  -245 - 84 t     (1)

6x + 7y  =  70  + 9t                (2)

Add (1)  and (2)

-50x = -175 - 75t

-2x = -7  - 3t

x =  (7 + 3t) / 2 =   (7/2) + (3/2)t

And

8(7/2 + (3/2)t )  + y  =  35 + 12t

28 + 12t + y  = 35 + 12t

y = 7 + 0t

The parameterized  equation  of the line  is

{ x , y , z}  =  ( 7/2 + (3/2)t , 7  , t }

And if this  line intersects the xy plane, then z =  t =  0

So.....the intersection point is  {x, y, z}  =   ( 7/2,  7, 0 )