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

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

Solve for x_1, x_2, x_3

$$x_{1}=\frac{11}{19}\approx 0.578947368$$

$$x_{2}=-\frac{9}{19}\approx -0.473684211$$

$$x_{3} = \frac{24}{19} = 1\frac{5}{19} \approx 1.263157895$$

Short Steps Using Substitution

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

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

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

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

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

$$x_{2}=-\frac{5}{7}x_{3}+\frac{3}{7} x_{3}=\frac{6}{7}-\frac{6}{7}x_{2}$$

Substitute -\frac{5}{7}x_{3}+\frac{3}{7} for x_{2} in the equation x_{3}=\frac{6}{7}-\frac{6}{7}x_{2}.

$$x_{3}=\frac{6}{7}-\frac{6}{7}\left(-\frac{5}{7}x_{3}+\frac{3}{7}\right)$$

Solve x_{3}=\frac{6}{7}-\frac{6}{7}\left(-\frac{5}{7}x_{3}+\frac{3}{7}\right) for x_{3}.

$$x_{3}=\frac{24}{19}$$

Substitute \frac{24}{19} for x_{3} in the equation x_{2}=-\frac{5}{7}x_{3}+\frac{3}{7}.

$$x_{2}=-\frac{5}{7}\times \frac{24}{19}+\frac{3}{7}$$

Calculate x_{2} from x_{2}=-\frac{5}{7}\times \frac{24}{19}+\frac{3}{7}.

$$x_{2}=-\frac{9}{19}$$

Substitute -\frac{9}{19} for x_{2} and \frac{24}{19} for x_{3} in the equation x_{1}=2x_{2}+2x_{3}-1.

$$x_{1}=2\left(-\frac{9}{19}\right)+2\times \frac{24}{19}-1$$

Calculate x_{1} from x_{1}=2\left(-\frac{9}{19}\right)+2\times \frac{24}{19}-1.

$$x_{1}=\frac{11}{19}$$

The system is now solved.

$$x_{1}=\frac{11}{19} x_{2}=-\frac{9}{19} x_{3}=\frac{24}{19}$$