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Math Help - locating a point closest to the origin using Lagrange multipliers?

  1. #1
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    locating a point closest to the origin using Lagrange multipliers?

    Here is the question...
    Locate the point on the line which is the intersection of the planes y+2z=12 and x+z=6 which is closest to the origin. Can someone solve this and please tell me how to do it using Lagrange multipliers? Thanks in advance.
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  2. #2
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    Quote Originally Posted by Infernorage View Post
    Here is the question...
    Locate the point on the line which is the intersection of the planes y+2z=12 and x+z=6 which is closest to the origin. Can someone solve this and please tell me how to do it using Lagrange multipliers? Thanks in advance.
    Let f(x, y, z) = x^2 + y^2 + z^2, g(x, y, z) = y + 2z, h(x, y, z) = x + z. Then you want to minimize f under the constraints g = 12 and h = 6. To do so, solve the system of five equations generated by grad(f) = lambda*grad(g) + mu*grad(h), g = 12, h = 6.
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  3. #3
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    Quote Originally Posted by rn443 View Post
    Let f(x, y, z) = x^2 + y^2 + z^2, g(x, y, z) = y + 2z, h(x, y, z) = x + z. Then you want to minimize f under the constraints g = 12 and h = 6. To do so, solve the system of five equations generated by grad(f) = lambda*grad(g) + mu*grad(h), g = 12, h = 6.
    Hi, yea I did that part. For the 5 equation I got the following...
    1) 2x=\mu
    2) 2y=\lambda
    3) 2z=2\lambda+\mu
    4) y+2z=12
    5) x+z=6
    I can't figure out to solve for the variables though. Can you help me out with this? Thanks.
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  4. #4
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    Quote Originally Posted by Infernorage View Post
    Hi, yea I did that part. For the 5 equation I got the following...
    1) 2x=\mu
    2) 2y=\lambda
    3) 2z=2\lambda+\mu
    4) y+2z=12
    5) x+z=6
    I can't figure out to solve for the variables though. Can you help me out with this? Thanks.
    Since \mu= 2x and \lambda= 2y, 2z= 2(2x)+ 2(2y) so z= 2x+ 2y. Putting that into y+ 2z= 12 gives y+ 2(2x+2y)= 4x+ 5y= 12.
    Putting z= 2x+ 2y into x+ z= 6 gives x+ (2x+2y)= 3x+ 2y= 6.

    Now, you have two equations to solve for x and y.
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