Solve {l}{2x_{1}+x_{2}+x_{3}=6}{3x_{1}-2x_{2}-3x_{3}=5}{8x_{1}+2x_{2}+5x_{3}=11}

$$\left. \begin{array} { l } { 2 x _ { 1 } + x _ { 2 } + x _ { 3 } = 6 } \\ { 3 x _ { 1 } - 2 x _ { 2 } - 3 x _ { 3 } = 5 } \\ { 8 x _ { 1 } + 2 x _ { 2 } + 5 x _ { 3 } = 11 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=2$$

$$x_{2}=5$$

$$x_{3}=-3$$

Short Steps Using Substitution

Solve 2x_{1}+x_{2}+x_{3}=6 for x_{2}.

$$x_{2}=-2x_{1}-x_{3}+6$$

Substitute -2x_{1}-x_{3}+6 for x_{2} in the second and third equation.

$$3x_{1}-2\left(-2x_{1}-x_{3}+6\right)-3x_{3}=5 8x_{1}+2\left(-2x_{1}-x_{3}+6\right)+5x_{3}=11$$

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

$$x_{1}=\frac{17}{7}+\frac{1}{7}x_{3} x_{3}=-\frac{4}{3}x_{1}-\frac{1}{3}$$

Substitute \frac{17}{7}+\frac{1}{7}x_{3} for x_{1} in the equation x_{3}=-\frac{4}{3}x_{1}-\frac{1}{3}.

$$x_{3}=-\frac{4}{3}\left(\frac{17}{7}+\frac{1}{7}x_{3}\right)-\frac{1}{3}$$

Solve x_{3}=-\frac{4}{3}\left(\frac{17}{7}+\frac{1}{7}x_{3}\right)-\frac{1}{3} for x_{3}.

$$x_{3}=-3$$

Substitute -3 for x_{3} in the equation x_{1}=\frac{17}{7}+\frac{1}{7}x_{3}.

$$x_{1}=\frac{17}{7}+\frac{1}{7}\left(-3\right)$$

Calculate x_{1} from x_{1}=\frac{17}{7}+\frac{1}{7}\left(-3\right).

$$x_{1}=2$$

Substitute 2 for x_{1} and -3 for x_{3} in the equation x_{2}=-2x_{1}-x_{3}+6.

$$x_{2}=-2\times 2-\left(-3\right)+6$$

Calculate x_{2} from x_{2}=-2\times 2-\left(-3\right)+6.

$$x_{2}=5$$

The system is now solved.

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