\left\{ \begin{array} { l } { 2 x _ { 1 } + x _ { 2 } - 3 x _ { 3 } = - 1 } \\ { x _ { 1 } - 3 x _ { 2 } + 2 x _ { 3 } = 10 } \\ { 3 x _ { 1 } - x _ { 2 } - x _ { 3 } = 5 } \end{array} \right.
Solve for x_1, x_2, x_3
x_{1}=0
x_{2}=-4
x_{3}=-1
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x_{2}=-2x_{1}+3x_{3}-1
Solve 2x_{1}+x_{2}-3x_{3}=-1 for x_{2}.
x_{1}-3\left(-2x_{1}+3x_{3}-1\right)+2x_{3}=10 3x_{1}-\left(-2x_{1}+3x_{3}-1\right)-x_{3}=5
Substitute -2x_{1}+3x_{3}-1 for x_{2} in the second and third equation.
x_{1}=1+x_{3} x_{3}=\frac{5}{4}x_{1}-1
Solve these equations for x_{1} and x_{3} respectively.
x_{3}=\frac{5}{4}\left(1+x_{3}\right)-1
Substitute 1+x_{3} for x_{1} in the equation x_{3}=\frac{5}{4}x_{1}-1.
x_{3}=-1
Solve x_{3}=\frac{5}{4}\left(1+x_{3}\right)-1 for x_{3}.
x_{1}=1-1
Substitute -1 for x_{3} in the equation x_{1}=1+x_{3}.
x_{1}=0
Calculate x_{1} from x_{1}=1-1.
x_{2}=-2\times 0+3\left(-1\right)-1
Substitute 0 for x_{1} and -1 for x_{3} in the equation x_{2}=-2x_{1}+3x_{3}-1.
x_{2}=-4
Calculate x_{2} from x_{2}=-2\times 0+3\left(-1\right)-1.
x_{1}=0 x_{2}=-4 x_{3}=-1
The system is now solved.
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