\left\{ \begin{array} { l } { x _ { 1 } + x _ { 2 } + x _ { 3 } = 3 } \\ { x _ { 1 } + 2 x _ { 2 } - x _ { 3 } = 2 } \\ { 4 x _ { 3 } + 7 x _ { 2 } - 2 x _ { 3 } = 9 } \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_{1}=-x_{2}-x_{3}+3
Solve x_{1}+x_{2}+x_{3}=3 for x_{1}.
-x_{2}-x_{3}+3+2x_{2}-x_{3}=2
Substitute -x_{2}-x_{3}+3 for x_{1} in the equation x_{1}+2x_{2}-x_{3}=2.
x_{2}=2x_{3}-1 x_{3}=-\frac{7}{2}x_{2}+\frac{9}{2}
Solve the second equation for x_{2} and the third equation for x_{3}.
x_{3}=-\frac{7}{2}\left(2x_{3}-1\right)+\frac{9}{2}
Substitute 2x_{3}-1 for x_{2} in the equation x_{3}=-\frac{7}{2}x_{2}+\frac{9}{2}.
x_{3}=1
Solve x_{3}=-\frac{7}{2}\left(2x_{3}-1\right)+\frac{9}{2} for x_{3}.
x_{2}=2\times 1-1
Substitute 1 for x_{3} in the equation x_{2}=2x_{3}-1.
x_{2}=1
Calculate x_{2} from x_{2}=2\times 1-1.
x_{1}=-1-1+3
Substitute 1 for x_{2} and 1 for x_{3} in the equation x_{1}=-x_{2}-x_{3}+3.
x_{1}=1
Calculate x_{1} from x_{1}=-1-1+3.
x_{1}=1 x_{2}=1 x_{3}=1
The system is now solved.
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