\left\{ \begin{array}{l}{ x _ { 1 } + x _ { 2 } + x _ { 3 } = 0 }\\{ x _ { 1 } + 2 x _ { 2 } - x _ { 3 } = 1 }\\{ 2 x _ { 1 } - 3 x _ { 2 } + x _ { 3 } = 2 }\end{array} \right.
Solve for x_1, x_2, x_3
x_{1}=\frac{10}{11}\approx 0.909090909
x_{2}=-\frac{3}{11}\approx -0.272727273
x_{3}=-\frac{7}{11}\approx -0.636363636
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x_{1}=-x_{2}-x_{3}
Solve x_{1}+x_{2}+x_{3}=0 for x_{1}.
-x_{2}-x_{3}+2x_{2}-x_{3}=1 2\left(-x_{2}-x_{3}\right)-3x_{2}+x_{3}=2
Substitute -x_{2}-x_{3} for x_{1} in the second and third equation.
x_{2}=2x_{3}+1 x_{3}=-5x_{2}-2
Solve these equations for x_{2} and x_{3} respectively.
x_{3}=-5\left(2x_{3}+1\right)-2
Substitute 2x_{3}+1 for x_{2} in the equation x_{3}=-5x_{2}-2.
x_{3}=-\frac{7}{11}
Solve x_{3}=-5\left(2x_{3}+1\right)-2 for x_{3}.
x_{2}=2\left(-\frac{7}{11}\right)+1
Substitute -\frac{7}{11} for x_{3} in the equation x_{2}=2x_{3}+1.
x_{2}=-\frac{3}{11}
Calculate x_{2} from x_{2}=2\left(-\frac{7}{11}\right)+1.
x_{1}=-\left(-\frac{3}{11}\right)-\left(-\frac{7}{11}\right)
Substitute -\frac{3}{11} for x_{2} and -\frac{7}{11} for x_{3} in the equation x_{1}=-x_{2}-x_{3}.
x_{1}=\frac{10}{11}
Calculate x_{1} from x_{1}=-\left(-\frac{3}{11}\right)-\left(-\frac{7}{11}\right).
x_{1}=\frac{10}{11} x_{2}=-\frac{3}{11} x_{3}=-\frac{7}{11}
The system is now solved.
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