Solve {l}{6=32x_{1}-10x_{2}-22x_{3}}{8=-10x_{1}+43x_{2}-33x_{3}}{0=-22x_{1}-33x_{2}+102x_{3}}

$$\left. \begin{array} { l } { 6 = 32 x _ { 1 } - 10 x _ { 2 } - 22 x _ { 3 } } \\ { 8 = - 10 x _ { 1 } + 43 x _ { 2 } - 33 x _ { 3 } } \\ { 0 = - 22 x _ { 1 } - 33 x _ { 2 } + 102 x _ { 3 } } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=\frac{16875}{29986}\approx 0.562762623$$

$$x_{2}=\frac{8179}{14993}\approx 0.545521243$$

$$x_{3}=\frac{14}{47}\approx 0.29787234$$

Short Steps Using Substitution

Solve 6=32x_{1}-10x_{2}-22x_{3} for x_{1}.

$$x_{1}=\frac{3}{16}+\frac{5}{16}x_{2}+\frac{11}{16}x_{3}$$

Substitute \frac{3}{16}+\frac{5}{16}x_{2}+\frac{11}{16}x_{3} for x_{1} in the second and third equation.

$$8=-10\left(\frac{3}{16}+\frac{5}{16}x_{2}+\frac{11}{16}x_{3}\right)+43x_{2}-33x_{3} 0=-22\left(\frac{3}{16}+\frac{5}{16}x_{2}+\frac{11}{16}x_{3}\right)-33x_{2}+102x_{3}$$

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

$$x_{2}=\frac{79}{319}+x_{3} x_{3}=\frac{33}{695}+\frac{319}{695}x_{2}$$

Substitute \frac{79}{319}+x_{3} for x_{2} in the equation x_{3}=\frac{33}{695}+\frac{319}{695}x_{2}.

$$x_{3}=\frac{33}{695}+\frac{319}{695}\left(\frac{79}{319}+x_{3}\right)$$

Solve x_{3}=\frac{33}{695}+\frac{319}{695}\left(\frac{79}{319}+x_{3}\right) for x_{3}.

$$x_{3}=\frac{14}{47}$$

Substitute \frac{14}{47} for x_{3} in the equation x_{2}=\frac{79}{319}+x_{3}.

$$x_{2}=\frac{79}{319}+\frac{14}{47}$$

Calculate x_{2} from x_{2}=\frac{79}{319}+\frac{14}{47}.

$$x_{2}=\frac{8179}{14993}$$

Substitute \frac{8179}{14993} for x_{2} and \frac{14}{47} for x_{3} in the equation x_{1}=\frac{3}{16}+\frac{5}{16}x_{2}+\frac{11}{16}x_{3}.

$$x_{1}=\frac{3}{16}+\frac{5}{16}\times \frac{8179}{14993}+\frac{11}{16}\times \frac{14}{47}$$

Calculate x_{1} from x_{1}=\frac{3}{16}+\frac{5}{16}\times \frac{8179}{14993}+\frac{11}{16}\times \frac{14}{47}.

$$x_{1}=\frac{16875}{29986}$$

The system is now solved.

$$x_{1}=\frac{16875}{29986} x_{2}=\frac{8179}{14993} x_{3}=\frac{14}{47}$$