Find the intersection of the planes. -7x - 4y - 8z = -123 and 6x - 8y - 4z = 42
A) x = 9 - 80t, y = -3 + 20t, z = 9 - 32t
B) x = 80t, y = -20t, z = 32t
C) x = 9 - 48t, y = -3 - 76t, z = 9 + 80t
D) The planes are parallel.

Question

Quizplus · Accepted Answer

We can solve for one variable in terms of the other two by eliminating one variable at a time using the equations of the planes.

From the first equation, we have: 
-7x - 4y - 8z = -123 
=> z = (7/8)x + (1/2)y + 123/8

Substituting this into the second equation: 
6x - 8y - 4z = 42 
=> 6x - 8y - 4((7/8)x + (1/2)y + 123/8) = 42 
=> (2/8)x - 5y - 123/2 = 42 
=> x = 9 + 48t, y = -3 - 76t, z = 9 + 80t

Therefore, the planes intersect at the point where x = 9 + 48t, y = -3 - 76t, and z = 9 + 80t, which matches choice C.

Find the Intersection of the Planes