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

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

Solve for x_1, x_2, x_3

$$x_{1}=-2$$

$$x_{2}=5$$

$$x_{3}=-3$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for x_{2} and x_{1} respectively.

$$x_{2}=1-2x_{1} x_{1}=\frac{14}{13}-\frac{8}{13}x_{2}$$

Substitute 1-2x_{1} for x_{2} in the equation x_{1}=\frac{14}{13}-\frac{8}{13}x_{2}.

$$x_{1}=\frac{14}{13}-\frac{8}{13}\left(1-2x_{1}\right)$$

Solve x_{1}=\frac{14}{13}-\frac{8}{13}\left(1-2x_{1}\right) for x_{1}.

$$x_{1}=-2$$

Substitute -2 for x_{1} in the equation x_{2}=1-2x_{1}.

$$x_{2}=1-2\left(-2\right)$$

Calculate x_{2} from x_{2}=1-2\left(-2\right).

$$x_{2}=5$$

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

$$x_{3}=5\left(-2\right)+2\times 5-3$$

Calculate x_{3} from x_{3}=5\left(-2\right)+2\times 5-3.

$$x_{3}=-3$$

The system is now solved.

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