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Question 29383: Hello, Im in a little bit of a pickle here...Our teacher gave us this question in class, and said we did not need to know anything more than what we already know. We havent learned matricies, or determinants or anything like that, but it seems to be the only way I can solve it. Is there way simpler?
The following system represents 3 planes which intersect in a line.
2x+3y-z=7
x-4y+5z=h
4x+ky+4z=8
Find the values of h and k.
(and does intersect on a line mean they are the same line...or something else?)
Thanks a lot!
Alex
Answer by longjonsilver(2297)   (Show Source):
You can put this solution on YOUR website! 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.
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