Solve {l}{3x_{1}+4x_{2}-2x_{3}=11}{2x_{1}-x_{2}-x_{3}=4}{3x_{1}-2x_{2}+4x_{3}=11}

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

Solve for x_1, x_2, x_3

$$x_{1}=3$$

$$x_{2}=1$$

$$x_{3}=1$$

Short Steps Using Substitution

Reorder the equations.

$$2x_{1}-x_{2}-x_{3}=4 3x_{1}+4x_{2}-2x_{3}=11 3x_{1}-2x_{2}+4x_{3}=11$$

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

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

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

$$3x_{1}+4\left(2x_{1}-x_{3}-4\right)-2x_{3}=11 3x_{1}-2\left(2x_{1}-x_{3}-4\right)+4x_{3}=11$$

Solve these equations for x_{1} and x_{3} respectively.

$$x_{1}=\frac{27}{11}+\frac{6}{11}x_{3} x_{3}=\frac{1}{6}x_{1}+\frac{1}{2}$$

Substitute \frac{27}{11}+\frac{6}{11}x_{3} for x_{1} in the equation x_{3}=\frac{1}{6}x_{1}+\frac{1}{2}.

$$x_{3}=\frac{1}{6}\left(\frac{27}{11}+\frac{6}{11}x_{3}\right)+\frac{1}{2}$$

Solve x_{3}=\frac{1}{6}\left(\frac{27}{11}+\frac{6}{11}x_{3}\right)+\frac{1}{2} for x_{3}.

$$x_{3}=1$$

Substitute 1 for x_{3} in the equation x_{1}=\frac{27}{11}+\frac{6}{11}x_{3}.

$$x_{1}=\frac{27}{11}+\frac{6}{11}\times 1$$

Calculate x_{1} from x_{1}=\frac{27}{11}+\frac{6}{11}\times 1.

$$x_{1}=3$$

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

$$x_{2}=2\times 3-1-4$$

Calculate x_{2} from x_{2}=2\times 3-1-4.

$$x_{2}=1$$

The system is now solved.

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