\left\{ \begin{array} { l } { 3 x _ { 1 } + 2 x _ { 2 } + x _ { 3 } = 5 } \\ { 2 x _ { 1 } + 3 x _ { 2 } + x _ { 3 } = 1 } \\ { 2 x _ { 1 } + x _ { 2 } + 3 x _ { 3 } = 11 } \end{array} \right.
Solve for x_1, x_2, x_3
x_{1}=2
x_{2}=-2
x_{3}=3
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x_{3}=-3x_{1}-2x_{2}+5
Solve 3x_{1}+2x_{2}+x_{3}=5 for x_{3}.
2x_{1}+3x_{2}-3x_{1}-2x_{2}+5=1 2x_{1}+x_{2}+3\left(-3x_{1}-2x_{2}+5\right)=11
Substitute -3x_{1}-2x_{2}+5 for x_{3} in the second and third equation.
x_{2}=x_{1}-4 x_{1}=-\frac{5}{7}x_{2}+\frac{4}{7}
Solve these equations for x_{2} and x_{1} respectively.
x_{1}=-\frac{5}{7}\left(x_{1}-4\right)+\frac{4}{7}
Substitute x_{1}-4 for x_{2} in the equation x_{1}=-\frac{5}{7}x_{2}+\frac{4}{7}.
x_{1}=2
Solve x_{1}=-\frac{5}{7}\left(x_{1}-4\right)+\frac{4}{7} for x_{1}.
x_{2}=2-4
Substitute 2 for x_{1} in the equation x_{2}=x_{1}-4.
x_{2}=-2
Calculate x_{2} from x_{2}=2-4.
x_{3}=-3\times 2-2\left(-2\right)+5
Substitute -2 for x_{2} and 2 for x_{1} in the equation x_{3}=-3x_{1}-2x_{2}+5.
x_{3}=3
Calculate x_{3} from x_{3}=-3\times 2-2\left(-2\right)+5.
x_{1}=2 x_{2}=-2 x_{3}=3
The system is now solved.
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