Solve {l}{2x_{1}+x_{2}-x_{3}=1}{x_{1}-3x_{1}+x_{3}=0}{x_{1}+4x_{2}-2x_{3}=2}

$$\left. \begin{array} { l } { 2 x _ { 1 } + x _ { 2 } - x _ { 3 } = 1 } \\ { x _ { 1 } - 3 x _ { 1 } + x _ { 3 } = 0 } \\ { x _ { 1 } + 4 x _ { 2 } - 2 x _ { 3 } = 2 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=\frac{2}{3}\approx 0.666666667$$

$$x_{2}=1$$

$$x_{3} = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Short Steps Using Substitution

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

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

Substitute -2x_{1}+x_{3}+1 for x_{2} in the equation x_{1}+4x_{2}-2x_{3}=2.

$$x_{1}+4\left(-2x_{1}+x_{3}+1\right)-2x_{3}=2$$

Solve the second equation for x_{1} and the third equation for x_{3}.

$$x_{1}=\frac{1}{2}x_{3} x_{3}=\frac{7}{2}x_{1}-1$$

Substitute \frac{1}{2}x_{3} for x_{1} in the equation x_{3}=\frac{7}{2}x_{1}-1.

$$x_{3}=\frac{7}{2}\times \frac{1}{2}x_{3}-1$$

Solve x_{3}=\frac{7}{2}\times \frac{1}{2}x_{3}-1 for x_{3}.

$$x_{3}=\frac{4}{3}$$

Substitute \frac{4}{3} for x_{3} in the equation x_{1}=\frac{1}{2}x_{3}.

$$x_{1}=\frac{1}{2}\times \frac{4}{3}$$

Calculate x_{1} from x_{1}=\frac{1}{2}\times \frac{4}{3}.

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

Substitute \frac{2}{3} for x_{1} and \frac{4}{3} for x_{3} in the equation x_{2}=-2x_{1}+x_{3}+1.

$$x_{2}=-2\times \frac{2}{3}+\frac{4}{3}+1$$

Calculate x_{2} from x_{2}=-2\times \frac{2}{3}+\frac{4}{3}+1.

$$x_{2}=1$$

The system is now solved.

$$x_{1}=\frac{2}{3} x_{2}=1 x_{3}=\frac{4}{3}$$