Solve for x_1, x_2, x_3
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x_{1}=-2x_{2}+x_{3}-3x_{4}
Solve x_{1}+2x_{2}-x_{3}+3x_{4}=0 for x_{1}.
2\left(-2x_{2}+x_{3}-3x_{4}\right)+3x_{2}-x_{3}+2x_{4}=0 -2x_{2}+x_{3}-3x_{4}+3x_{3}+3x_{4}=0
Substitute -2x_{2}+x_{3}-3x_{4} for x_{1} in the second and third equation.
x_{2}=x_{3}-4x_{4} x_{3}=\frac{1}{2}x_{2}
Solve these equations for x_{2} and x_{3} respectively.
x_{3}=\frac{1}{2}\left(x_{3}-4x_{4}\right)
Substitute x_{3}-4x_{4} for x_{2} in the equation x_{3}=\frac{1}{2}x_{2}.
x_{3}=-4x_{4}
Solve x_{3}=\frac{1}{2}\left(x_{3}-4x_{4}\right) for x_{3}.
x_{2}=-4x_{4}-4x_{4}
Substitute -4x_{4} for x_{3} in the equation x_{2}=x_{3}-4x_{4}.
x_{2}=-8x_{4}
Calculate x_{2} from x_{2}=-4x_{4}-4x_{4}.
x_{1}=-2\left(-8\right)x_{4}-4x_{4}-3x_{4}
Substitute -8x_{4} for x_{2} and -4x_{4} for x_{3} in the equation x_{1}=-2x_{2}+x_{3}-3x_{4}.
x_{1}=9x_{4}
Calculate x_{1} from x_{1}=-2\left(-8\right)x_{4}-4x_{4}-3x_{4}.
x_{1}=9x_{4} x_{2}=-8x_{4} x_{3}=-4x_{4}
The system is now solved.