Solve {r}{x_{1}-5x_{2}+2x_{3}=-20}{-3x_{1}+x_{2}-x_{3}=8}{-2x_{2}+5x_{3}=-16}

$$\left. \begin{array} { r } { x _ { 1 } - 5 x _ { 2 } + 2 x _ { 3 } = - 20 } \\ { - 3 x _ { 1 } + x _ { 2 } - x _ { 3 } = 8 } \\ { - 2 x _ { 2 } + 5 x _ { 3 } = - 16 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=-1$$

$$x_{2}=3$$

$$x_{3}=-2$$

Short Steps Using Substitution

Solve x_{1}-5x_{2}+2x_{3}=-20 for x_{1}.

$$x_{1}=5x_{2}-2x_{3}-20$$

Substitute 5x_{2}-2x_{3}-20 for x_{1} in the equation -3x_{1}+x_{2}-x_{3}=8.

$$-3\left(5x_{2}-2x_{3}-20\right)+x_{2}-x_{3}=8$$

Solve the second equation for x_{2} and the third equation for x_{3}.

$$x_{2}=\frac{5}{14}x_{3}+\frac{26}{7} x_{3}=-\frac{16}{5}+\frac{2}{5}x_{2}$$

Substitute \frac{5}{14}x_{3}+\frac{26}{7} for x_{2} in the equation x_{3}=-\frac{16}{5}+\frac{2}{5}x_{2}.

$$x_{3}=-\frac{16}{5}+\frac{2}{5}\left(\frac{5}{14}x_{3}+\frac{26}{7}\right)$$

Solve x_{3}=-\frac{16}{5}+\frac{2}{5}\left(\frac{5}{14}x_{3}+\frac{26}{7}\right) for x_{3}.

$$x_{3}=-2$$

Substitute -2 for x_{3} in the equation x_{2}=\frac{5}{14}x_{3}+\frac{26}{7}.

$$x_{2}=\frac{5}{14}\left(-2\right)+\frac{26}{7}$$

Calculate x_{2} from x_{2}=\frac{5}{14}\left(-2\right)+\frac{26}{7}.

$$x_{2}=3$$

Substitute 3 for x_{2} and -2 for x_{3} in the equation x_{1}=5x_{2}-2x_{3}-20.

$$x_{1}=5\times 3-2\left(-2\right)-20$$

Calculate x_{1} from x_{1}=5\times 3-2\left(-2\right)-20.

$$x_{1}=-1$$

The system is now solved.

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