Solve {l}{3x_{1}-2x_{2}=3}{-4x_{1}+3x_{2}=-1}{x_{1}+2x_{2}+x_{3}=0}

$$\left\{ \begin{array} { l } { 3 x _ { 1 } - 2 x _ { 2 } = 3 } \\ { - 4 x _ { 1 } + 3 x _ { 2 } = - 1 } \\ { x _ { 1 } + 2 x _ { 2 } + x _ { 3 } = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=7$$

$$x_{2}=9$$

$$x_{3}=-25$$

Short Steps Using Substitution

Reorder the equations.

$$x_{1}+2x_{2}+x_{3}=0 -4x_{1}+3x_{2}=-1 3x_{1}-2x_{2}=3$$

Solve x_{1}+2x_{2}+x_{3}=0 for x_{1}.

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

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

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

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

$$x_{2}=-\frac{4}{11}x_{3}-\frac{1}{11} x_{3}=-1-\frac{8}{3}x_{2}$$

Substitute -\frac{4}{11}x_{3}-\frac{1}{11} for x_{2} in the equation x_{3}=-1-\frac{8}{3}x_{2}.

$$x_{3}=-1-\frac{8}{3}\left(-\frac{4}{11}x_{3}-\frac{1}{11}\right)$$

Solve x_{3}=-1-\frac{8}{3}\left(-\frac{4}{11}x_{3}-\frac{1}{11}\right) for x_{3}.

$$x_{3}=-25$$

Substitute -25 for x_{3} in the equation x_{2}=-\frac{4}{11}x_{3}-\frac{1}{11}.

$$x_{2}=-\frac{4}{11}\left(-25\right)-\frac{1}{11}$$

Calculate x_{2} from x_{2}=-\frac{4}{11}\left(-25\right)-\frac{1}{11}.

$$x_{2}=9$$

Substitute 9 for x_{2} and -25 for x_{3} in the equation x_{1}=-2x_{2}-x_{3}.

$$x_{1}=-2\times 9-\left(-25\right)$$

Calculate x_{1} from x_{1}=-2\times 9-\left(-25\right).

$$x_{1}=7$$

The system is now solved.

$$x_{1}=7 x_{2}=9 x_{3}=-25$$