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

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

Solve for x_2, x_3, x_1

$$x_{2}=\frac{1}{17}\approx 0.058823529$$

$$x_{3} = \frac{45}{17} = 2\frac{11}{17} \approx 2.647058824$$

$$x_{1} = -\frac{93}{17} = -5\frac{8}{17} \approx -5.470588235$$

Short Steps Using Substitution

Solve x_{2}+3x_{3}=8 for x_{2}.

$$x_{2}=-3x_{3}+8$$

Substitute -3x_{3}+8 for x_{2} in the second and third equation.

$$4x_{1}+6\left(-3x_{3}+8\right)+7x_{3}=-3 2x_{1}-3x_{3}+8+6x_{3}=5$$

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

$$x_{3}=\frac{4}{11}x_{1}+\frac{51}{11} x_{1}=-\frac{3}{2}-\frac{3}{2}x_{3}$$

Substitute \frac{4}{11}x_{1}+\frac{51}{11} for x_{3} in the equation x_{1}=-\frac{3}{2}-\frac{3}{2}x_{3}.

$$x_{1}=-\frac{3}{2}-\frac{3}{2}\left(\frac{4}{11}x_{1}+\frac{51}{11}\right)$$

Solve x_{1}=-\frac{3}{2}-\frac{3}{2}\left(\frac{4}{11}x_{1}+\frac{51}{11}\right) for x_{1}.

$$x_{1}=-\frac{93}{17}$$

Substitute -\frac{93}{17} for x_{1} in the equation x_{3}=\frac{4}{11}x_{1}+\frac{51}{11}.

$$x_{3}=\frac{4}{11}\left(-\frac{93}{17}\right)+\frac{51}{11}$$

Calculate x_{3} from x_{3}=\frac{4}{11}\left(-\frac{93}{17}\right)+\frac{51}{11}.

$$x_{3}=\frac{45}{17}$$

Substitute \frac{45}{17} for x_{3} in the equation x_{2}=-3x_{3}+8.

$$x_{2}=-3\times \frac{45}{17}+8$$

Calculate x_{2} from x_{2}=-3\times \frac{45}{17}+8.

$$x_{2}=\frac{1}{17}$$

The system is now solved.

$$x_{2}=\frac{1}{17} x_{3}=\frac{45}{17} x_{1}=-\frac{93}{17}$$