Solve {l}{2x_{1}-6x_{2}-13=-38}{-3x_{1}-x_{2}+7x_{3}=-34}{-8x_{1}+x_{2}-2x_{2}=-20}

$$\left. \begin{array}{l}{ 2 x _ { 1 } - 6 x _ { 2 } - 13 = - 38 }\\{ - 3 x _ { 1 } - x _ { 2 } + 7 x _ { 3 } = - 34 }\\{ - 8 x _ { 1 } + x _ { 2 } - 2 x _ { 2 } = - 20 }\end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{19}{10} = 1\frac{9}{10} = 1.9$$

$$x_{2} = \frac{24}{5} = 4\frac{4}{5} = 4.8$$

$$x_{3} = -\frac{47}{14} = -3\frac{5}{14} \approx -3.357142857$$

Short Steps Using Substitution

Reorder the equations.

$$-3x_{1}-x_{2}+7x_{3}=-34 2x_{1}-6x_{2}-13=-38 -8x_{1}+x_{2}-2x_{2}=-20$$

Solve -3x_{1}-x_{2}+7x_{3}=-34 for x_{2}.

$$x_{2}=-3x_{1}+7x_{3}+34$$

Substitute -3x_{1}+7x_{3}+34 for x_{2} in the second and third equation.

$$2x_{1}-6\left(-3x_{1}+7x_{3}+34\right)-13=-38 -8x_{1}-3x_{1}+7x_{3}+34-2\left(-3x_{1}+7x_{3}+34\right)=-20$$

Solve these equations for x_{1} and x_{3} respectively.

$$x_{1}=\frac{179}{20}+\frac{21}{10}x_{3} x_{3}=-2-\frac{5}{7}x_{1}$$

Substitute \frac{179}{20}+\frac{21}{10}x_{3} for x_{1} in the equation x_{3}=-2-\frac{5}{7}x_{1}.

$$x_{3}=-2-\frac{5}{7}\left(\frac{179}{20}+\frac{21}{10}x_{3}\right)$$

Solve x_{3}=-2-\frac{5}{7}\left(\frac{179}{20}+\frac{21}{10}x_{3}\right) for x_{3}.

$$x_{3}=-\frac{47}{14}$$

Substitute -\frac{47}{14} for x_{3} in the equation x_{1}=\frac{179}{20}+\frac{21}{10}x_{3}.

$$x_{1}=\frac{179}{20}+\frac{21}{10}\left(-\frac{47}{14}\right)$$

Calculate x_{1} from x_{1}=\frac{179}{20}+\frac{21}{10}\left(-\frac{47}{14}\right).

$$x_{1}=\frac{19}{10}$$

Substitute \frac{19}{10} for x_{1} and -\frac{47}{14} for x_{3} in the equation x_{2}=-3x_{1}+7x_{3}+34.

$$x_{2}=-3\times \frac{19}{10}+7\left(-\frac{47}{14}\right)+34$$

Calculate x_{2} from x_{2}=-3\times \frac{19}{10}+7\left(-\frac{47}{14}\right)+34.

$$x_{2}=\frac{24}{5}$$

The system is now solved.

$$x_{1}=\frac{19}{10} x_{2}=\frac{24}{5} x_{3}=-\frac{47}{14}$$