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Thread: Sysyem

  1. #1
    Super Member dhiab's Avatar
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    Sysyem

    Solve the system in R :
    x + y = 1
    y + z = 1
    x + z = 1
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  2. #2
    Junior Member
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    Guideline

    express y as function of x
    put it in second equation
    express z as function of x
    put in third equation
    find z
    put z in second equation
    find y
    put y in first equation
    find x
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  3. #3
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    Hello, dhiab!

    Solve the system in R:

    . . $\displaystyle \begin{array}{cccccccc}
    x & + & y & & & = & 1 & [1] \\
    & & y & + & z & = & 1 & [2] \\
    x & & & + &z & = & 1 & [3] \end{array}$

    $\displaystyle \begin{array}{ccccc}\text{Subtract [1]-[2]:} & x - z &=& 0 \\
    \text{Add [3]:} & x + z &=& 1 \end{array}$

    And we have: .$\displaystyle 2x \:=\:1 \quad\Rightarrow\quad \boxed{x \:=\:\tfrac{1}{2}}$

    Substitute into [1]: .$\displaystyle \tfrac{1}{2} + y \:=\:1 \quad\Rightarrow\quad\boxed{ y \:=\:\tfrac{1}{2}}$

    Substitute into [2]: .$\displaystyle \tfrac{1}{2} + z \:=\:1 \quad\Rightarrow\quad \boxed{z \:=\:\tfrac{1}{2}}$

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  4. #4
    MHF Contributor red_dog's Avatar
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    Another approach:

    Add all the equations: $\displaystyle 2(x+y+z)=3\Rightarrow x+y+z=\frac{3}{2}$.

    $\displaystyle x+y=1\Rightarrow z=\frac{1}{2}$

    $\displaystyle y+z=1\Rightarrow x=\frac{1}{2}$

    $\displaystyle x+z=1\Rightarrow y=\frac{1}{2}$
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