Solve {l}{x_{1}-5x_{2}-2x_{3}=4}{2x_{1}-3x_{2}+x_{3}=7}{-x_{1}+12x_{2}+7x_{1}=-5}

$$\left\{ \begin{array} { l } { x _ { 1 } - 5 x _ { 2 } - 2 x _ { 3 } = 4 } \\ { 2 x _ { 1 } - 3 x _ { 2 } + x _ { 3 } = 7 } \\ { - x _ { 1 } + 12 x _ { 2 } + 7 x _ { 1 } = - 5 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{23}{18} = 1\frac{5}{18} \approx 1.277777778$$

$$x_{2} = -\frac{19}{18} = -1\frac{1}{18} \approx -1.055555556$$

$$x_{3} = \frac{23}{18} = 1\frac{5}{18} \approx 1.277777778$$

Short Steps Using Substitution

Solve x_{1}-5x_{2}-2x_{3}=4 for x_{1}.

$$x_{1}=5x_{2}+2x_{3}+4$$

Substitute 5x_{2}+2x_{3}+4 for x_{1} in the second and third equation.

$$2\left(5x_{2}+2x_{3}+4\right)-3x_{2}+x_{3}=7 -\left(5x_{2}+2x_{3}+4\right)+12x_{2}+7\left(5x_{2}+2x_{3}+4\right)=-5$$

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

$$x_{2}=-\frac{5}{7}x_{3}-\frac{1}{7} x_{3}=-\frac{29}{12}-\frac{7}{2}x_{2}$$

Substitute -\frac{5}{7}x_{3}-\frac{1}{7} for x_{2} in the equation x_{3}=-\frac{29}{12}-\frac{7}{2}x_{2}.

$$x_{3}=-\frac{29}{12}-\frac{7}{2}\left(-\frac{5}{7}x_{3}-\frac{1}{7}\right)$$

Solve x_{3}=-\frac{29}{12}-\frac{7}{2}\left(-\frac{5}{7}x_{3}-\frac{1}{7}\right) for x_{3}.

$$x_{3}=\frac{23}{18}$$

Substitute \frac{23}{18} for x_{3} in the equation x_{2}=-\frac{5}{7}x_{3}-\frac{1}{7}.

$$x_{2}=-\frac{5}{7}\times \frac{23}{18}-\frac{1}{7}$$

Calculate x_{2} from x_{2}=-\frac{5}{7}\times \frac{23}{18}-\frac{1}{7}.

$$x_{2}=-\frac{19}{18}$$

Substitute -\frac{19}{18} for x_{2} and \frac{23}{18} for x_{3} in the equation x_{1}=5x_{2}+2x_{3}+4.

$$x_{1}=5\left(-\frac{19}{18}\right)+2\times \frac{23}{18}+4$$

Calculate x_{1} from x_{1}=5\left(-\frac{19}{18}\right)+2\times \frac{23}{18}+4.

$$x_{1}=\frac{23}{18}$$

The system is now solved.

$$x_{1}=\frac{23}{18} x_{2}=-\frac{19}{18} x_{3}=\frac{23}{18}$$