Find the point at which the line intersects the given plane.
x = 4 − t, y = 5 + t, z = 2t;
x − y + 3z = 3
(x, y, z) =____________?
see
http://www.jiskha.com/display.cgi?id=1391198824
for the method
To find the point at which the line intersects the given plane, we need to find the values of x, y, and z that satisfy both the equations of the line and the equation of the plane.
Given:
Line equations: x = 4 − t, y = 5 + t, z = 2t
Plane equation: x − y + 3z = 3
Step 1: Substitute the line equations into the plane equation.
(4 − t) − (5 + t) + 3(2t) = 3
Step 2: Simplify the equation.
4 − t − 5 − t + 6t = 3
-2t + 4t = 4
2t = 4
t = 2
Step 3: Substitute the value of t back into the line equations to find x, y, and z.
x = 4 − t = 4 − 2 = 2
y = 5 + t = 5 + 2 = 7
z = 2t = 2(2) = 4
Therefore, the point of intersection between the line and the plane is (2, 7, 4).