Solve {l}{x_{1}+x_{2}-x_{3}=0}{2x_{1}-x_{3}=2}{4x_{1}+x_{2}+x_{3}=6}

$$\left. \begin{array}{l}{ x _ { 1 } + x _ { 2 } - x _ { 3 } = 0 }\\{ 2 x _ { 1 } - x _ { 3 } = 2 }\\{ 4 x _ { 1 } + x _ { 2 } + x _ { 3 } = 6 }\end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{10}{7} = 1\frac{3}{7} \approx 1.428571429$$

$$x_{2}=-\frac{4}{7}\approx -0.571428571$$

$$x_{3}=\frac{6}{7}\approx 0.857142857$$

Short Steps Using Substitution

Solve x_{1}+x_{2}-x_{3}=0 for x_{1}.

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

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

$$2\left(-x_{2}+x_{3}\right)-x_{3}=2 4\left(-x_{2}+x_{3}\right)+x_{2}+x_{3}=6$$

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

$$x_{2}=-1+\frac{1}{2}x_{3} x_{3}=\frac{6}{5}+\frac{3}{5}x_{2}$$

Substitute -1+\frac{1}{2}x_{3} for x_{2} in the equation x_{3}=\frac{6}{5}+\frac{3}{5}x_{2}.

$$x_{3}=\frac{6}{5}+\frac{3}{5}\left(-1+\frac{1}{2}x_{3}\right)$$

Solve x_{3}=\frac{6}{5}+\frac{3}{5}\left(-1+\frac{1}{2}x_{3}\right) for x_{3}.

$$x_{3}=\frac{6}{7}$$

Substitute \frac{6}{7} for x_{3} in the equation x_{2}=-1+\frac{1}{2}x_{3}.

$$x_{2}=-1+\frac{1}{2}\times \frac{6}{7}$$

Calculate x_{2} from x_{2}=-1+\frac{1}{2}\times \frac{6}{7}.

$$x_{2}=-\frac{4}{7}$$

Substitute -\frac{4}{7} for x_{2} and \frac{6}{7} for x_{3} in the equation x_{1}=-x_{2}+x_{3}.

$$x_{1}=-\left(-\frac{4}{7}\right)+\frac{6}{7}$$

Calculate x_{1} from x_{1}=-\left(-\frac{4}{7}\right)+\frac{6}{7}.

$$x_{1}=\frac{10}{7}$$

The system is now solved.

$$x_{1}=\frac{10}{7} x_{2}=-\frac{4}{7} x_{3}=\frac{6}{7}$$