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Aufgaben zum Gaußverfahren

  1. 1

    Löse das Gleichungssystem mit dem Gauß-Jordan-Verfahren.

    1. 3x+4y=12x+5y=3\def\arraystretch{1.25} \begin{array}{ccccc}3x&+&4y&=&-1\\2x&+&5y&=&-3\end{array}

    2. 3x4y=262x+3y=28\def\arraystretch{1.25} \begin{array}{ccrcc}3x&-&4y&=&-26\\2x&+&3y&=&28\end{array}

    3.     x+2yz=2x+y+2z=92x+3y3z=1\def\arraystretch{1.25} \begin{array}{rcrcrcr}\;\;x&+&2y&-&z&=&2\\x&+&y&+&2z&=&9\\2x&+&3y&-&3z&=&-1\end{array}

  2. 2

    Löse das Gleichungssystem mit dem Gaußverfahren.

    1. 3x+2y=14x+y=26x+4y=3\def\arraystretch{1.25} \begin{array}{ccrcr}3x&+&2y&=&-1\\4x&+&y&=&-2\\6x&+&4y&=&3\end{array}

    2.   x3y=42x+y=14x+5y=9\def\arraystretch{1.25} \begin{array}{rcrcc}\;x&-&3y&=&4\\2x&+&y&=&1\\4x&+&5y&=&9\end{array}

    3. x2y=32x+4y=6  x+2y=3\def\arraystretch{1.25} \begin{array}{rcrcr}x&-&2y&=&3\\-2x&+&4y&=&-6\\\;-x&+&2y&=&-3\end{array}

    4. 6xy+2z=15x3y+3z=43x2y+z=14\def\arraystretch{1.25} \begin{array}{rcrcrcr}6x&-&y&+&2z&=&1\\5x&-&3y&+&3z&=&4\\3x&-&2y&+&z&=&14\end{array}

    5. 34x76y=189x+14y=3213x+19y=0\def\arraystretch{1.25} \begin{array}{cccc}\frac34x&-&\frac76y&=&\frac18\\-9x&+&14y&=&-\frac32\\\frac13x&+&\frac19y&=&0\end{array}

    6. 6xz+2y=485y3x+3z=493z2x+y=24\def\arraystretch{1.25} \begin{array}{rcrcrcr}6x &- & z &+ &2y &=&48\\5y &- &3x &+ &3z &=&49\\3z &- &2x &+ & y &=&24\end{array}

    7.     x2y    =4  yz=1x+y+3z=1\def\arraystretch{1.25} \begin{array}{rcrcrcr}\;\;x&-&2y&\;&\;&=&4\\\;&-&y&-&z&=&-1\\-x&+&y&+&3z&=&-1\end{array}

    8. 2x+3yz=3x    +2z=9xy    =2\def\arraystretch{1.25} \begin{array}{rcrcrcr}2x&+&3y&-&z&=&3\\x&\;&\;&+&2z&=&9\\x&-&y&\;&\;&=&2\end{array}

    9. 5x+2y2z=13x+y3z=42x    +z=4\def\arraystretch{1.25} \begin{array}{rcrcrcr}5x&+&2y&-&2z&=&-1\\3x&+&y&-&3z&=&-4\\2x&\;&\;&+&z&=&4\end{array}

    10. 4x+3y+z=132x5y+3z=17xy2z=1\def\arraystretch{1.25} \begin{array}{ccrcrcr}4x&+&3y&+&z&=&13\\2x&-&5y&+&3z&=&1\\7x&-&y&-&2z&=&-1\end{array}

    11. 2x+9y14z=393x+6y+2z=36x2+y3+7z=2\def\arraystretch{1.25} \begin{array}{rcrcrcr}2x&+&9y&-&14z&=&39\\3x&+&6y&+&2z&=&36\\\frac x2&+&\frac y3&+&7z&=&2\end{array}