Leysa {l}{3x_{1}+x_{2}-7x_{3}=17}{4x_{1}+5x_{2}=1}{-2x_{1}+11x_{3}=-24}

$$\left. \begin{array}{l}{ 3 x _ { 1 } + x _ { 2 } - 7 x _ { 3 } = 17 }\\{ 4 x _ { 1 } + 5 x _ { 2 } = 1 }\\{ - 2 x _ { 1 } + 11 x _ { 3 } = - 24 }\end{array} \right.$$

Leystu fyrir x_1, x_2, x_3

$$x_{1} = \frac{28}{17} = 1\frac{11}{17} \approx 1.647058824$$

$$x_{2} = -\frac{19}{17} = -1\frac{2}{17} \approx -1.117647059$$

$$x_{3} = -\frac{32}{17} = -1\frac{15}{17} \approx -1.882352941$$

Styttri leiðir við notkun innsetningar

Leystu 3x_{1}+x_{2}-7x_{3}=17 fyrir x_{2}.

$$x_{2}=-3x_{1}+7x_{3}+17$$

Settu -3x_{1}+7x_{3}+17 inn fyrir x_{2} í hinni jöfnunni 4x_{1}+5x_{2}=1.

$$4x_{1}+5\left(-3x_{1}+7x_{3}+17\right)=1$$

Leysa aðra jöfnu fyrir x_{1} og þá þriðju fyrir x_{3}.

$$x_{1}=\frac{84}{11}+\frac{35}{11}x_{3} x_{3}=-\frac{24}{11}+\frac{2}{11}x_{1}$$

Settu \frac{84}{11}+\frac{35}{11}x_{3} inn fyrir x_{1} í hinni jöfnunni x_{3}=-\frac{24}{11}+\frac{2}{11}x_{1}.

$$x_{3}=-\frac{24}{11}+\frac{2}{11}\left(\frac{84}{11}+\frac{35}{11}x_{3}\right)$$

Leystu x_{3}=-\frac{24}{11}+\frac{2}{11}\left(\frac{84}{11}+\frac{35}{11}x_{3}\right) fyrir x_{3}.

$$x_{3}=-\frac{32}{17}$$

Settu -\frac{32}{17} inn fyrir x_{3} í hinni jöfnunni x_{1}=\frac{84}{11}+\frac{35}{11}x_{3}.

$$x_{1}=\frac{84}{11}+\frac{35}{11}\left(-\frac{32}{17}\right)$$

Reiknaðu x_{1} frá x_{1}=\frac{84}{11}+\frac{35}{11}\left(-\frac{32}{17}\right).

$$x_{1}=\frac{28}{17}$$

Settu \frac{28}{17} inn fyrir x_{1} og -\frac{32}{17} fyrir x_{3} í jöfnunni x_{2}=-3x_{1}+7x_{3}+17.

$$x_{2}=-3\times \frac{28}{17}+7\left(-\frac{32}{17}\right)+17$$

Reiknaðu x_{2} frá x_{2}=-3\times \frac{28}{17}+7\left(-\frac{32}{17}\right)+17.

$$x_{2}=-\frac{19}{17}$$

Leyst var úr kerfinu.

$$x_{1}=\frac{28}{17} x_{2}=-\frac{19}{17} x_{3}=-\frac{32}{17}$$