Solve {l}{x_{1}-x_{2}+2x_{3}=-2}{-3x_{1}+6x_{2}-8x_{3}=13}{5x_{1}+4x_{2}+6x_{3}=7}

$$\left. \begin{array}{l}{ x _ { 1 } - x _ { 2 } + 2 x _ { 3 } = - 2 }\\{ - 3 x _ { 1 } + 6 x _ { 2 } - 8 x _ { 3 } = 13 }\\{ 5 x _ { 1 } + 4 x _ { 2 } + 6 x _ { 3 } = 7 }\end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=3$$

$$x_{2}=1$$

$$x_{3}=-2$$

Short Steps Using Substitution

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

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

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

$$-3\left(x_{2}-2x_{3}-2\right)+6x_{2}-8x_{3}=13 5\left(x_{2}-2x_{3}-2\right)+4x_{2}+6x_{3}=7$$

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

$$x_{2}=\frac{2}{3}x_{3}+\frac{7}{3} x_{3}=\frac{9}{4}x_{2}-\frac{17}{4}$$

Substitute \frac{2}{3}x_{3}+\frac{7}{3} for x_{2} in the equation x_{3}=\frac{9}{4}x_{2}-\frac{17}{4}.

$$x_{3}=\frac{9}{4}\left(\frac{2}{3}x_{3}+\frac{7}{3}\right)-\frac{17}{4}$$

Solve x_{3}=\frac{9}{4}\left(\frac{2}{3}x_{3}+\frac{7}{3}\right)-\frac{17}{4} for x_{3}.

$$x_{3}=-2$$

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

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

Calculate x_{2} from x_{2}=\frac{2}{3}\left(-2\right)+\frac{7}{3}.

$$x_{2}=1$$

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

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

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

$$x_{1}=3$$

The system is now solved.

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