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