Answer:
(11/3, 26/3, -9)
Step-by-step explanation:
You want the point of intersection of the plane 2x +y +z = 7 and the line through points (2, -3, 1) and (3, 4, -5).
Line
The line through points A and B can be written in parametric form as ...
P = A(1 -t) +B(t)
P = (2(1 -t) +3t, -3(1 -t) +4t, 1(1 -t) -5t)
P = (t +2, 7t -3, 1 -6t)
Intersection
The value of t where this hits the plane can be found by using P in the equation of the plane:
2(t +2) +(7t -3) +(1 -6t) = 7
3t = 5 . . . . . . . . . . . . . . . . . . simplify, subtract 2
t = 5/3
The point P for this value of t is ...
P = (5/3 +2, 7(5/3) -3 1 -6(5/3)) = (3 2/3, 8 2/3, -9)
The point where the line cuts the plane is (11/3, 26/3, -9).
