Solve {l}{-2x_{1}+3x_{2}-x_{3}=7}{4x_{1}-2x_{2}+5x_{3}=-3text{es}}{x_{1}+2x_{2}-3x_{3}=0}

$$\left\{ \begin{array} { l } { - 2 x _ { 1 } + 3 x _ { 2 } - x _ { 3 } = 7 } \\ { 4 x _ { 1 } - 2 x _ { 2 } + 5 x _ { 3 } = - 3 \text { es } } \\ { x _ { 1 } + 2 x _ { 2 } - 3 x _ { 3 } = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=\frac{-3es-4}{7}$$

$$x_{2}=\frac{17-3es}{7}$$

$$x_{3}=\frac{10-3es}{7}$$

Short Steps Using Substitution

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

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

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

$$4x_{1}-2x_{2}+5\left(-2x_{1}+3x_{2}-7\right)=-3es x_{1}+2x_{2}-3\left(-2x_{1}+3x_{2}-7\right)=0$$

Solve these equations for x_{2} and x_{1} respectively.

$$x_{2}=\frac{6}{13}x_{1}+\frac{35}{13}-\frac{3}{13}es x_{1}=-3+x_{2}$$

Substitute \frac{6}{13}x_{1}+\frac{35}{13}-\frac{3}{13}es for x_{2} in the equation x_{1}=-3+x_{2}.

$$x_{1}=-3+\frac{6}{13}x_{1}+\frac{35}{13}-\frac{3}{13}es$$

Solve x_{1}=-3+\frac{6}{13}x_{1}+\frac{35}{13}-\frac{3}{13}es for x_{1}.

$$x_{1}=-\frac{3}{7}es-\frac{4}{7}$$

Substitute -\frac{3}{7}es-\frac{4}{7} for x_{1} in the equation x_{2}=\frac{6}{13}x_{1}+\frac{35}{13}-\frac{3}{13}es.

$$x_{2}=\frac{6}{13}\left(-\frac{3}{7}es-\frac{4}{7}\right)+\frac{35}{13}-\frac{3}{13}es$$

Calculate x_{2} from x_{2}=\frac{6}{13}\left(-\frac{3}{7}es-\frac{4}{7}\right)+\frac{35}{13}-\frac{3}{13}es.

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

Substitute -\frac{3}{7}se+\frac{17}{7} for x_{2} and -\frac{3}{7}es-\frac{4}{7} for x_{1} in the equation x_{3}=-2x_{1}+3x_{2}-7.

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

Calculate x_{3} from x_{3}=-2\left(-\frac{3}{7}es-\frac{4}{7}\right)+3\left(-\frac{3}{7}se+\frac{17}{7}\right)-7.

$$x_{3}=\frac{10}{7}-\frac{3}{7}se$$

The system is now solved.

$$x_{1}=-\frac{3}{7}es-\frac{4}{7} x_{2}=-\frac{3}{7}se+\frac{17}{7} x_{3}=\frac{10}{7}-\frac{3}{7}se$$