Final answer:
The equations of the line are given by x=-3-4t, y=-1+2t, z=3+4t. After substituting these into the plane equation 2x + 2y - 2z = -86 and solving, we get the solution t=-7. Substituting t=-7 into the line equations results in the intersection point (25, -15, -25).
Step-by-step explanation:
To find the point of intersection of a line and a plane we can substitute the parametric equations of the line into the equation of the plane. Here we have a line ⟨-3, -1, 3⟩ t ⟨-4, 2, 4⟩. These can be written as equations of the line in terms of t as follows: x = -3 - 4t, y = -1 + 2t, and z = 3 + 4t. We can substitute these into the equation for the plane:
2(-3 - 4t) + 2(-1 + 2t) - 2(3 + 4t) = -86
This eventually simplifies to t = -7. So, substituting t = -7 into the line equations gives the point of intersection as (25, -15, -25).
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