Solve {l}{3x_{1}+x_{2}-7x_{3}=17}{4x_{1}+5x_{2}=1}{-2x_{1}+11x_{3}=-24}

$$\left. \begin{array}{l}{ 3 x _ { 1 } + x _ { 2 } - 7 x _ { 3 } = 17 }\\{ 4 x _ { 1 } + 5 x _ { 2 } = 1 }\\{ - 2 x _ { 1 } + 11 x _ { 3 } = - 24 }\end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{28}{17} = 1\frac{11}{17} \approx 1.647058824$$

$$x_{2} = -\frac{19}{17} = -1\frac{2}{17} \approx -1.117647059$$

$$x_{3} = -\frac{32}{17} = -1\frac{15}{17} \approx -1.882352941$$

Short Steps Using Substitution

Solve 3x_{1}+x_{2}-7x_{3}=17 for x_{2}.

$$x_{2}=-3x_{1}+7x_{3}+17$$

Substitute -3x_{1}+7x_{3}+17 for x_{2} in the equation 4x_{1}+5x_{2}=1.

$$4x_{1}+5\left(-3x_{1}+7x_{3}+17\right)=1$$

Solve the second equation for x_{1} and the third equation for x_{3}.

$$x_{1}=\frac{84}{11}+\frac{35}{11}x_{3} x_{3}=-\frac{24}{11}+\frac{2}{11}x_{1}$$

Substitute \frac{84}{11}+\frac{35}{11}x_{3} for x_{1} in the equation x_{3}=-\frac{24}{11}+\frac{2}{11}x_{1}.

$$x_{3}=-\frac{24}{11}+\frac{2}{11}\left(\frac{84}{11}+\frac{35}{11}x_{3}\right)$$

Solve x_{3}=-\frac{24}{11}+\frac{2}{11}\left(\frac{84}{11}+\frac{35}{11}x_{3}\right) for x_{3}.

$$x_{3}=-\frac{32}{17}$$

Substitute -\frac{32}{17} for x_{3} in the equation x_{1}=\frac{84}{11}+\frac{35}{11}x_{3}.

$$x_{1}=\frac{84}{11}+\frac{35}{11}\left(-\frac{32}{17}\right)$$

Calculate x_{1} from x_{1}=\frac{84}{11}+\frac{35}{11}\left(-\frac{32}{17}\right).

$$x_{1}=\frac{28}{17}$$

Substitute \frac{28}{17} for x_{1} and -\frac{32}{17} for x_{3} in the equation x_{2}=-3x_{1}+7x_{3}+17.

$$x_{2}=-3\times \frac{28}{17}+7\left(-\frac{32}{17}\right)+17$$

Calculate x_{2} from x_{2}=-3\times \frac{28}{17}+7\left(-\frac{32}{17}\right)+17.

$$x_{2}=-\frac{19}{17}$$

The system is now solved.

$$x_{1}=\frac{28}{17} x_{2}=-\frac{19}{17} x_{3}=-\frac{32}{17}$$