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Solve for x_1, x_2, x_3
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x_{2}=2x_{1}-x_{3}
Solve 2x_{1}-x_{2}-x_{3}=0 for x_{2}.
11-\left(2x_{1}-x_{3}\right)-3x_{3}=-12 114x_{1}-\left(2x_{1}-x_{3}\right)+2x_{3}=15
Substitute 2x_{1}-x_{3} for x_{2} in the second and third equation.
x_{1}=\frac{23}{2}-x_{3} x_{3}=-\frac{112}{3}x_{1}+5
Solve these equations for x_{1} and x_{3} respectively.
x_{3}=-\frac{112}{3}\left(\frac{23}{2}-x_{3}\right)+5
Substitute \frac{23}{2}-x_{3} for x_{1} in the equation x_{3}=-\frac{112}{3}x_{1}+5.
x_{3}=\frac{1273}{109}
Solve x_{3}=-\frac{112}{3}\left(\frac{23}{2}-x_{3}\right)+5 for x_{3}.
x_{1}=\frac{23}{2}-\frac{1273}{109}
Substitute \frac{1273}{109} for x_{3} in the equation x_{1}=\frac{23}{2}-x_{3}.
x_{1}=-\frac{39}{218}
Calculate x_{1} from x_{1}=\frac{23}{2}-\frac{1273}{109}.
x_{2}=2\left(-\frac{39}{218}\right)-\frac{1273}{109}
Substitute -\frac{39}{218} for x_{1} and \frac{1273}{109} for x_{3} in the equation x_{2}=2x_{1}-x_{3}.
x_{2}=-\frac{1312}{109}
Calculate x_{2} from x_{2}=2\left(-\frac{39}{218}\right)-\frac{1273}{109}.
x_{1}=-\frac{39}{218} x_{2}=-\frac{1312}{109} x_{3}=\frac{1273}{109}
The system is now solved.