Determine the line through which the planes in each pair intersect.
a) 3x+2y+5z=4
4x-3y+z=-1
2 answers
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1st times 3: --> 9x + 6y + 15z = 12
2nd times 2: --> 8x - 6y + 2z = -2
add them: 17x + 17z = 10 , we wanted to eliminate one of the variables
x+z = 10/17
Pick any value of z to get z
let z=0, then x = 10/17
in#2: 40/17 - 3y + 0 = -1
-3y = -57/17
y = 19/17 -------->a point is (10/17, 19/17, 0)
let x = 0 , then z = 10/17
in #2: 0 - 3y + 10/17 = -1
-3y = -27/17
y = 9/17 --------> a point (0, 9/17 , 10/17)
no we have two points on our line, not "nice" points, but hey ....
direction vector: (10/17 - 0 , 19/17 - 9/17 , 0 - 10/17)
= (10/17, 0 , -10/17)
or we could just use (10, 10, -10)
or even better:
(1, 1, -1)
so using the point (10/17, 19/17,0)
we have the parametric equation:
x = 10/17 + t
y = 19/17 + t
z = -t
be aware that this equation is not unique , but the direction vector must be a multiple of (1,1,-1)
2nd times 2: --> 8x - 6y + 2z = -2
add them: 17x + 17z = 10 , we wanted to eliminate one of the variables
x+z = 10/17
Pick any value of z to get z
let z=0, then x = 10/17
in#2: 40/17 - 3y + 0 = -1
-3y = -57/17
y = 19/17 -------->a point is (10/17, 19/17, 0)
let x = 0 , then z = 10/17
in #2: 0 - 3y + 10/17 = -1
-3y = -27/17
y = 9/17 --------> a point (0, 9/17 , 10/17)
no we have two points on our line, not "nice" points, but hey ....
direction vector: (10/17 - 0 , 19/17 - 9/17 , 0 - 10/17)
= (10/17, 0 , -10/17)
or we could just use (10, 10, -10)
or even better:
(1, 1, -1)
so using the point (10/17, 19/17,0)
we have the parametric equation:
x = 10/17 + t
y = 19/17 + t
z = -t
be aware that this equation is not unique , but the direction vector must be a multiple of (1,1,-1)
1 answer