Solve {l}{x_{1}+0,06x_{2}-0,02x_{3}=2}{0,03x_{1}+x_{2}-0,05x_{3}=3}{0,01x_{1}-0,02x_{2}+x_{3}=5}

$$\left\{ \begin{array} { l } { x _ { 1 } + 0,06 x _ { 2 } - 0,02 x _ { 3 } = 2 } \\ { 0,03 x _ { 1 } + x _ { 2 } - 0,05 x _ { 3 } = 3 } \\ { 0,01 x _ { 1 } - 0,02 x _ { 2 } + x _ { 3 } = 5 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=\frac{952100}{498691}\approx 1,909198281$$

$$x_{2}=\frac{1593300}{498691}\approx 3,194964417$$

$$x_{3}=\frac{2515800}{498691}\approx 5,044807306$$

Short Steps Using Substitution

Solve x_{1}+0,06x_{2}-0,02x_{3}=2 for x_{1}.

$$x_{1}=-0,06x_{2}+0,02x_{3}+2$$

Substitute -0,06x_{2}+0,02x_{3}+2 for x_{1} in the second and third equation.

$$0,03\left(-0,06x_{2}+0,02x_{3}+2\right)+x_{2}-0,05x_{3}=3 0,01\left(-0,06x_{2}+0,02x_{3}+2\right)-0,02x_{2}+x_{3}=5$$

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

$$x_{2}=\frac{247}{4991}x_{3}+\frac{2100}{713} x_{3}=\frac{103}{5001}x_{2}+\frac{8300}{1667}$$

Substitute \frac{247}{4991}x_{3}+\frac{2100}{713} for x_{2} in the equation x_{3}=\frac{103}{5001}x_{2}+\frac{8300}{1667}.

$$x_{3}=\frac{103}{5001}\left(\frac{247}{4991}x_{3}+\frac{2100}{713}\right)+\frac{8300}{1667}$$

Solve x_{3}=\frac{103}{5001}\left(\frac{247}{4991}x_{3}+\frac{2100}{713}\right)+\frac{8300}{1667} for x_{3}.

$$x_{3}=\frac{2515800}{498691}$$

Substitute \frac{2515800}{498691} for x_{3} in the equation x_{2}=\frac{247}{4991}x_{3}+\frac{2100}{713}.

$$x_{2}=\frac{247}{4991}\times \frac{2515800}{498691}+\frac{2100}{713}$$

Calculate x_{2} from x_{2}=\frac{247}{4991}\times \frac{2515800}{498691}+\frac{2100}{713}.

$$x_{2}=\frac{1593300}{498691}$$

Substitute \frac{1593300}{498691} for x_{2} and \frac{2515800}{498691} for x_{3} in the equation x_{1}=-0,06x_{2}+0,02x_{3}+2.

$$x_{1}=-0,06\times \frac{1593300}{498691}+0,02\times \frac{2515800}{498691}+2$$

Calculate x_{1} from x_{1}=-0,06\times \frac{1593300}{498691}+0,02\times \frac{2515800}{498691}+2.

$$x_{1}=\frac{952100}{498691}$$

The system is now solved.

$$x_{1}=\frac{952100}{498691} x_{2}=\frac{1593300}{498691} x_{3}=\frac{2515800}{498691}$$