Question

Find the point at which the line intersects the given plane.
x = 1 + 4t, y = 4t, z = 2−3t ;
x + 2y − z + 1 = 0

(x, y, z) = _____________?

Answer 1

substitute the parameterization of each component, x, y, and z, into the equation of the plane and solve for t.
then plug that t back into the 3 equations to determine the values of x, y, and z

Answer 2

x = 1 + 4t
y = 4t
z = 2 - 3t
now sub these back into the other equation
x + 2y - z + 1 = 0
(1 + 4t) + 2(4t) - (2 - 3t) + 1 = 0
1 + 4t + 8t - 2 + 3t + 1 = 0 --- combine like terms
15t = 0
t = 0
now sub 0 in for t in the other equations
x = 1 + 4t
x = 1 + 4(0)
x = 1 + 0
x = 1

y = 4t
y = 4(0)
y = 0

z = 2 - 3t
z = 2 - 3(0)
z = 2 - 0
z = 2

check...
x + 2y - z + 1 = 0
1 + 2(0) - 2 + 1 = 0
1 - 2 + 1 = 0
2 - 2 = 0
0 = 0 (correct)

x = 1, y = 0, and z = 2
any questions ?

Answer 3

thank you