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Solutions to an equation

  1. May 5, 2013 #1
    1. The problem statement, all variables and given/known data
    If in an equation |x|+|y|+|z|=10, x,y,z ##\in## I, then the number of solutions are
    A)528
    B)402
    C)666
    D)None


    2. Relevant equations



    3. The attempt at a solution
    I am clueless on this one. It looks to me that it represents 8 planes but I don't think that's going to help me here.
     
  2. jcsd
  3. May 5, 2013 #2
    All values of [itex]x,y,z[/itex] are within [itex][-10,10][/itex], in order for that equation to be satisfied. How many different ways can you add 3 integers ([itex]x,y,z[/itex] s.t. [itex]0 \leq x,y,z \leq 10[/itex]) to be equal to 10. Don't worry about negatives for now.
     
  4. May 5, 2013 #3
    For 0<= x,y,z <=10, the number of solutions are 12C2. I tried this before but then got stuck when it came to my mind that I need to consider the negatives too.
     
  5. May 5, 2013 #4
    How many solutions do you have when you have [itex]0 \leq y,z \leq 10[/itex], and [itex]-10 \leq x \leq 0[/itex]? I would iterate your process fixing one, then 2, then 3 variables to be negative. You could then setup an equation that adds each of those solution sets together and subtract duplicate solutions.
     
  6. May 5, 2013 #5

    haruspex

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    Yes, it's 8 planes, specifically the surface of a regular octahedron. So you can use symmetry. This is the same as Coto suggests, but using a more geometric (visual) approach. E.g. count one plane, multiply by 8, then set about subtracting the double counts: each edge was counted twice, each vertex ... how many times?
     
  7. May 6, 2013 #6
    During the examination, I used the same approach but I did not end up with the right answer.

    Considering the plane x+y+z=10, the integer solutions where x,y,z>0 are 9C2. Multiplying by 8 and adding up the edges, I get 9C2*8+10*6+1 (+1 for the origin)=349 which is incorrect. I don't see how this is wrong.

    EDIT:
    Considering the plane x+y+z=10, the integer solutions where x,y,z>0 are 9C2. We have 12 edges. The number of points on those edges are (not including the vertices) 9*12 and adding up the vertices i.e. 6
    I get 9C2*8+12*9+6=402. Woops, looks like my approach was correct, did some silly mistakes in the exam. -_-

    I looked at the given solution, it is given that required solutions=9C1*3C1*2^2+9C2*2^3+1*3C2*2=402. Can you explain me why did they write 3C1 and 2^2?

    Thank you haruspex and Coto! :smile:
     
    Last edited: May 6, 2013
  8. May 7, 2013 #7

    haruspex

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    This is for number of solutions along edges of the octahedron, excluding the vertices. The are 9 such along each edge, so you're asking why they write 3C1*2^2 for the number of edges. An edge corresponds to one of the variables being 0: 3C1. The remaining two variables are nonzero and can have either sign independently: 2^2.
     
  9. May 7, 2013 #8
    Got it, thank you haruspex! :smile:
     
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