\left\{ \begin{array}{l}{ x _ { 1 } - 2 x _ { 2 } + 3 x _ { 3 } = 6 }\\{ 2 x _ { 1 } + 3 x _ { 2 } - 4 x _ { 3 } = 16 }\\{ 3 x _ { 1 } - 2 x _ { 2 } - 5 x _ { 3 } = 12 }\end{array} \right.
Solve for x_1, x_2, x_3
x_{1}=7
x_{2}=2
x_{3}=1
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x_{1}=2x_{2}-3x_{3}+6
Solve x_{1}-2x_{2}+3x_{3}=6 for x_{1}.
2\left(2x_{2}-3x_{3}+6\right)+3x_{2}-4x_{3}=16 3\left(2x_{2}-3x_{3}+6\right)-2x_{2}-5x_{3}=12
Substitute 2x_{2}-3x_{3}+6 for x_{1} in the second and third equation.
x_{2}=\frac{4}{7}+\frac{10}{7}x_{3} x_{3}=\frac{3}{7}+\frac{2}{7}x_{2}
Solve these equations for x_{2} and x_{3} respectively.
x_{3}=\frac{3}{7}+\frac{2}{7}\left(\frac{4}{7}+\frac{10}{7}x_{3}\right)
Substitute \frac{4}{7}+\frac{10}{7}x_{3} for x_{2} in the equation x_{3}=\frac{3}{7}+\frac{2}{7}x_{2}.
x_{3}=1
Solve x_{3}=\frac{3}{7}+\frac{2}{7}\left(\frac{4}{7}+\frac{10}{7}x_{3}\right) for x_{3}.
x_{2}=\frac{4}{7}+\frac{10}{7}\times 1
Substitute 1 for x_{3} in the equation x_{2}=\frac{4}{7}+\frac{10}{7}x_{3}.
x_{2}=2
Calculate x_{2} from x_{2}=\frac{4}{7}+\frac{10}{7}\times 1.
x_{1}=2\times 2-3+6
Substitute 2 for x_{2} and 1 for x_{3} in the equation x_{1}=2x_{2}-3x_{3}+6.
x_{1}=7
Calculate x_{1} from x_{1}=2\times 2-3+6.
x_{1}=7 x_{2}=2 x_{3}=1
The system is now solved.
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