Solve {l}{x_{1}-x_{2}+x_{3}=6}{2x_{1}-x_{2}+3x_{3}=4}{4x_{1}+5x_{2}-10x_{3}=13}

$$\left. \begin{array} { l } { x _ { 1 } - x _ { 2 } + x _ { 3 } = 6 } \\ { 2 x _ { 1 } - x _ { 2 } + 3 x _ { 3 } = 4 } \\ { 4 x _ { 1 } + 5 x _ { 2 } - 10 x _ { 3 } = 13 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{76}{23} = 3\frac{7}{23} \approx 3.304347826$$

$$x_{2} = -\frac{123}{23} = -5\frac{8}{23} \approx -5.347826087$$

$$x_{3} = -\frac{61}{23} = -2\frac{15}{23} \approx -2.652173913$$

Short Steps Using Substitution

Solve x_{1}-x_{2}+x_{3}=6 for x_{1}.

$$x_{1}=x_{2}-x_{3}+6$$

Substitute x_{2}-x_{3}+6 for x_{1} in the second and third equation.

$$2\left(x_{2}-x_{3}+6\right)-x_{2}+3x_{3}=4 4\left(x_{2}-x_{3}+6\right)+5x_{2}-10x_{3}=13$$

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

$$x_{2}=-x_{3}-8 x_{3}=\frac{9}{14}x_{2}+\frac{11}{14}$$

Substitute -x_{3}-8 for x_{2} in the equation x_{3}=\frac{9}{14}x_{2}+\frac{11}{14}.

$$x_{3}=\frac{9}{14}\left(-x_{3}-8\right)+\frac{11}{14}$$

Solve x_{3}=\frac{9}{14}\left(-x_{3}-8\right)+\frac{11}{14} for x_{3}.

$$x_{3}=-\frac{61}{23}$$

Substitute -\frac{61}{23} for x_{3} in the equation x_{2}=-x_{3}-8.

$$x_{2}=-\left(-\frac{61}{23}\right)-8$$

Calculate x_{2} from x_{2}=-\left(-\frac{61}{23}\right)-8.

$$x_{2}=-\frac{123}{23}$$

Substitute -\frac{123}{23} for x_{2} and -\frac{61}{23} for x_{3} in the equation x_{1}=x_{2}-x_{3}+6.

$$x_{1}=-\frac{123}{23}-\left(-\frac{61}{23}\right)+6$$

Calculate x_{1} from x_{1}=-\frac{123}{23}-\left(-\frac{61}{23}\right)+6.

$$x_{1}=\frac{76}{23}$$

The system is now solved.

$$x_{1}=\frac{76}{23} x_{2}=-\frac{123}{23} x_{3}=-\frac{61}{23}$$