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