Find the equation of the plane that passes through the line of intersection of the planes x - 3y - 2z - 1 = 0 and 2x + 4y + z - 5 = 0 and parallel to the x-axis.

Find the line of intersection for the two planes:
1: 0t 3𝑦 − 2𝑧 − 1 = 0
2: 2𝑥 + 4𝑦 + 𝑧 − 5 = 0
let 𝑥=𝑡 (a scalar)
0𝑡+3𝑦 − 2𝑧 − 1 = 0(1)
2𝑡 + 4𝑦 + 𝑧 − 5 = 0..(2)
use substitution to find the value of y from equation (1):
0𝑡+3𝑦 − 2𝑧 − 1 = 0..(1)
3𝑦= 2𝑧+ 1
𝑦=2𝑧+13(3)
eliminate the variable y from the equations:
First Multiply equation (1) by 4, and multiply equation (2) by -3:
0𝑡+12𝑦 −8𝑧 − 4 = 0
−6𝑡−12𝑦−3𝑧+15 = 0
Use the elimination method:
0𝑡+12𝑦 −8𝑧 − 4 = 0
−6𝑡−12𝑦−3𝑧+15 = 0
−6𝑡−11𝑧+11=0
−6𝑡−11𝑧=−11(4)
−11𝑧=−11+6𝑡
𝑧=−6𝑡/11+1
y:
Substitute the value of z into equation (3):
𝑦=2𝑧+13
𝑦=2(−6𝑡/11+1)
𝑦=(−4𝑡−11)/11

I am very confused by the decimal values...Does my working make sense? I would appreciate any help

2. Originally Posted by spoc21 Find the equation of the plane that passes through the line of intersection of the planes x - 3y - 2z - 1 = 0 and 2x + 4y + z - 5 = 0 and parallel to the x-axis.

Find the line of intersection for the two planes:
1: 0t 3�� − 2�� − 1 = 0
2: 2�� + 4�� + �� − 5 = 0
let ��=�� (a scalar)
0��+3�� − 2�� − 1 = 0(1)
2�� + 4�� + �� − 5 = 0..(2)
use substitution to find the value of y from equation (1):
0��+3�� − 2�� − 1 = 0..(1)
3��= 2��+ 1
��=2��+13(3)
eliminate the variable y from the equations:
First Multiply equation (1) by 4, and multiply equation (2) by -3:
0��+12�� −8�� − 4 = 0
−6��−12��−3��+15 = 0
Use the elimination method:
0��+12�� −8�� − 4 = 0
−6��−12��−3��+15 = 0
−6��−11��+11=0
−6��−11��=−11(4)
−11��=−11+6��
��=−6��/11+1
y:
Substitute the value of z into equation (3):
��=2��+13
��=2(−6��/11+1)
��=(−4��−11)/11

I am very confused by the decimal values...Does my working make sense? I would appreciate any help
Any equation of a plane passing through the intersection of 2 planes is given by (x-3y-2z-1)+(lambda)(2x+4y+z-5)=0 where lambda=l be any constant.Simplifying we get x(1+2l)+y(4l-3)+z(l-2)-6l=0-------(1).But it is given that the resulting plane is parallel to the x-axis. therefore the coefficient of y in (1) must be 0. thus 4l-3=0 =>l=3/4. Putting this value in (1) we should the value of the required plane which according to my calculations is:10x-5z-18=0. Hope this was helpful!!!

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