Solve {l}{x_{1}+x_{2}+x_{3}=1}{-x_{1}+x_{2}+x_{3}=-1}{x_{1}-x_{2}+2x_{3}=0}

$$\left. \begin{array} { l } { x _ { 1 } + x _ { 2 } + x _ { 3 } = 1 } \\ { - x _ { 1 } + x _ { 2 } + x _ { 3 } = - 1 } \\ { x _ { 1 } - x _ { 2 } + 2 x _ { 3 } = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=1$$

$$x_{2}=\frac{1}{3}\approx 0.333333333$$

$$x_{3}=-\frac{1}{3}\approx -0.333333333$$

Short Steps Using Substitution

Solve x_{1}+x_{2}+x_{3}=1 for x_{1}.

$$x_{1}=-x_{2}-x_{3}+1$$

Substitute -x_{2}-x_{3}+1 for x_{1} in the second and third equation.

$$-\left(-x_{2}-x_{3}+1\right)+x_{2}+x_{3}=-1 -x_{2}-x_{3}+1-x_{2}+2x_{3}=0$$

Solve these equations for x_{2} and x_{3} respectively.

$$x_{2}=-x_{3} x_{3}=2x_{2}-1$$

Substitute -x_{3} for x_{2} in the equation x_{3}=2x_{2}-1.

$$x_{3}=2\left(-1\right)x_{3}-1$$

Solve x_{3}=2\left(-1\right)x_{3}-1 for x_{3}.

$$x_{3}=-\frac{1}{3}$$

Substitute -\frac{1}{3} for x_{3} in the equation x_{2}=-x_{3}.

$$x_{2}=-\left(-\frac{1}{3}\right)$$

Calculate x_{2} from x_{2}=-\left(-\frac{1}{3}\right).

$$x_{2}=\frac{1}{3}$$

Substitute \frac{1}{3} for x_{2} and -\frac{1}{3} for x_{3} in the equation x_{1}=-x_{2}-x_{3}+1.

$$x_{1}=-\frac{1}{3}-\left(-\frac{1}{3}\right)+1$$

Calculate x_{1} from x_{1}=-\frac{1}{3}-\left(-\frac{1}{3}\right)+1.

$$x_{1}=1$$

The system is now solved.

$$x_{1}=1 x_{2}=\frac{1}{3} x_{3}=-\frac{1}{3}$$