Solve {c}{x_{1}-5x_{2}+4x_{3}=-7}{2x_{1}-9x_{2}-x_{3}=4}{3x_{1}-11x_{2}-7x_{3}=17}

$$\left\{ \begin{array} { c } { x _ { 1 } - 5 x _ { 2 } + 4 x _ { 3 } = - 7 } \\ { 2 x _ { 1 } - 9 x _ { 2 } - x _ { 3 } = 4 } \\ { 3 x _ { 1 } - 11 x _ { 2 } - 7 x _ { 3 } = 17 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=1$$

$$x_{2}=0$$

$$x_{3}=-2$$

Short Steps Using Substitution

Solve x_{1}-5x_{2}+4x_{3}=-7 for x_{1}.

$$x_{1}=5x_{2}-4x_{3}-7$$

Substitute 5x_{2}-4x_{3}-7 for x_{1} in the second and third equation.

$$2\left(5x_{2}-4x_{3}-7\right)-9x_{2}-x_{3}=4 3\left(5x_{2}-4x_{3}-7\right)-11x_{2}-7x_{3}=17$$

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

$$x_{2}=18+9x_{3} x_{3}=-2+\frac{4}{19}x_{2}$$

Substitute 18+9x_{3} for x_{2} in the equation x_{3}=-2+\frac{4}{19}x_{2}.

$$x_{3}=-2+\frac{4}{19}\left(18+9x_{3}\right)$$

Solve x_{3}=-2+\frac{4}{19}\left(18+9x_{3}\right) for x_{3}.

$$x_{3}=-2$$

Substitute -2 for x_{3} in the equation x_{2}=18+9x_{3}.

$$x_{2}=18+9\left(-2\right)$$

Calculate x_{2} from x_{2}=18+9\left(-2\right).

$$x_{2}=0$$

Substitute 0 for x_{2} and -2 for x_{3} in the equation x_{1}=5x_{2}-4x_{3}-7.

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

Calculate x_{1} from x_{1}=5\times 0-4\left(-2\right)-7.

$$x_{1}=1$$

The system is now solved.

$$x_{1}=1 x_{2}=0 x_{3}=-2$$