Question 29383
imagine a table top being one plane and a sheet of metal lands in the table top. That is the second plane. Imagine a second sheet of metal that also lands in the table along the same entry line as the first sheet.


You have 3 planes that intersect along a line...meaning there are many "answers", "solutions" to the 3 equations.


2x+3y-z=7
x-4y+5z=h
4x+ky+4z=8


So, take 2 equations and find where they are equal. I shall use
2x+3y-z=7 and x-4y+5z=h. I shall scale the sexond equation up by factor of 2 to make the x-terms the same:


2x+3y-z=7
2x-8y+10z=2h


Subtracting, we have
11y-11z = 7-2h --> eqn1


Now looking at 2x+3y-z=7 and 4x+ky+4z=8, and scaling the first up by factor of 2 to give:


4x+6y-2z=14
4x+ky+4z=8


Subtract these to give: (6-k)y-6z=6. Now this equation has to be same equation as eqn1, since it is where te 2 planes intersect...the same line on the table if you will, so we need to figure out the values of h and k from:


11y-11z = 7-2h --> x6
(6-k)y-6z=6   --> x11


66y - 66z = 42-12h
(66-11k)y - 66z = 66


So, looking at the y term, the coefficient 66 and (66-11k) must be the same:
66 = 66-11k
11k = 0 
--> k = 0


And also 66 = 42-12h
--> 24 = -12h
--> h = 24/-12
--> h = -2


I believe the 3 equations are:
2x+3y-z=7
x-4y+5z=-2
4x+4z=8


Please check the working.


jon.