Results 1 to 3 of 3
Like Tree2Thanks
  • 2 Post By Soroban

Math Help - Math Problem help

  1. #1
    Member
    Joined
    Oct 2012
    From
    san francisco
    Posts
    92

    Math Problem help

    Hey there sorry if this is in the wrong thread, but I need some help with this problem

    If x^2+y^2+z^2=49, and x+y+z=x^3+y^3+z^3=7
    find xyz

    I've been trying this problem for a while now and was hoping if someone can guide me through this problem so I can understand it better thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,689
    Thanks
    617

    Re: Math Problem help

    Hello, gfbrd!

    I'm familiar with this type of problem,
    but this one took me a couple of tries . . .


    \text{Given: }\:\begin{Bmatrix}x+y+z \:=\:7 & [1] \\ x^2+y^2+z^2 \:=\:49 & [2] \\ x^3+y^3 +z^3 \:=\:7 & [3] \end{Bmatrix}

    \text{Find }xyz.

    Square [1]: . (x+y+z)^2 \:=\:7^2

    . . x^2 + 2xy + 2xz + y^2 + 2xz + z^2 \:=\:49

    . . 2(xy + yz + xz) + \underbrace{(x^2+y^2+z^2)}_{\text{This is 49}} \:=\:49

    Hence: . 2(xy+yz+xz) \:=\:0 \quad\Rightarrow\quad xy + yz + xz \:=\:0\;\;[4]


    Cube [1]: . (x+y)^3 \:=\:7^3

    . . \begin{bmatrix}x^3 \\ 3x^2y + 3x^2z \\ 3xy^2 + 6xyz + 3xz^2 \\ y^3 + 3y^2z + 3yz^2 + z^3 \end{bmatrix} \;=\;343


    Add 3xyz to both sides:

    . . \begin{bmatrix}x^3 \\ 3x^2y + 3x^2z \\ 3xy^2 + 9xyz + 3xz^2 \\ y^3 + 3y^2z + 3yz^2 + z^3 \end{bmatrix} \;=\;343 + 3xyz


    The left side is:

    . . \overbrace{(x^3 + y^3 + z^3)}^{\text{This is 7}} + (3x^2y + 3x^2y + 3xyz) + (3y^2z + 3yz^2 + 3xyz)
    . . . . + (3x^2z + 3xz^2 + 3xyz) \;=\; 343 + 3xyz

    . . 7 + 3xy(x+y+z) + 3yz(x+y+z) + 3xz(x+y+z) \;=\;343 + 3xyz

    . . 7 + 3\underbrace{(xy + yz + xz)}_{\text{This is 0}}\underbrace{(x+y+z)}_{\text{This is 7}} \;=\;343 + 3xyz


    Therefore: .  7 \;=\;343 + 3xyz \quad\Rightarrow\quad 3xyz \:=\:-336 \quad\Rightarrow\quad \boxed{xyz \:=\:-112}
    Thanks from gfbrd and HallsofIvy
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2012
    From
    san francisco
    Posts
    92

    Re: Math Problem help

    Nice work soroban, I was working on this for so long and had trouble with it, thank you so much.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. GRE Math problem help!
    Posted in the Geometry Forum
    Replies: 1
    Last Post: October 7th 2010, 02:13 PM
  2. Help with math problem
    Posted in the Algebra Forum
    Replies: 2
    Last Post: October 18th 2009, 09:19 AM
  3. Help with math problem
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: September 21st 2009, 03:12 PM
  4. I need help with math problem...
    Posted in the Math Topics Forum
    Replies: 5
    Last Post: September 22nd 2008, 05:19 PM
  5. math problem!
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 3rd 2008, 07:04 PM

Search Tags


/mathhelpforum @mathhelpforum