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