Final answer:
The point of intersection between the given line and plane is (9/4, -4, 3/2).
Step-by-step explanation:
The line is defined by the equations:
x = 3 + 5t
y = -4
z = 2t
The equation of the plane is 2x + 3y - 2z = 6.
To find the point of intersection, we can substitute the values of x, y, and z from the line equations into the plane equation:
2(3 + 5t) + 3(-4) - 2(2t) = 6
Simplifying this equation gives:
6 + 10t - 12 + 6t = 6
16t = 12
t = 12/16
t = 3/4
Substituting t back into the line equations gives:
x = 3 + 5(3/4)
y = -4
z = 2(3/4)
Simplifying these equations gives:
x = 9/4
y = -4
z = 3/2
Therefore, the point of intersection is (9/4, -4, 3/2).