Solve {l}{0.7x_{1}+0.1x_{2}+0.05x_{3}=1}{0.15x_{1}+0.8x_{2}+0.05x_{3}=1}{x_{1}+x_{2}+x_{3}=1}

$$\left. \begin{array} { l } { 0.7 x _ { 1 } + 0.1 x _ { 2 } + 0.05 x _ { 3 } = 1 } \\ { 0.15 x _ { 1 } + 0.8 x _ { 2 } + 0.05 x _ { 3 } = 1 } \\ { x _ { 1 } + x _ { 2 } + x _ { 3 } = 1 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{266}{193} = 1\frac{73}{193} \approx 1.378238342$$

$$x_{2} = \frac{209}{193} = 1\frac{16}{193} \approx 1.082901554$$

$$x_{3} = -\frac{282}{193} = -1\frac{89}{193} \approx -1.461139896$$

Short Steps Using Substitution

Reorder the equations.

$$x_{1}+x_{2}+x_{3}=1 0.15x_{1}+0.8x_{2}+0.05x_{3}=1 0.7x_{1}+0.1x_{2}+0.05x_{3}=1$$

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

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

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

$$0.15\left(-x_{2}-x_{3}+1\right)+0.8x_{2}+0.05x_{3}=1 0.7\left(-x_{2}-x_{3}+1\right)+0.1x_{2}+0.05x_{3}=1$$

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

$$x_{2}=\frac{2}{13}x_{3}+\frac{17}{13} x_{3}=-\frac{12}{13}x_{2}-\frac{6}{13}$$

Substitute \frac{2}{13}x_{3}+\frac{17}{13} for x_{2} in the equation x_{3}=-\frac{12}{13}x_{2}-\frac{6}{13}.

$$x_{3}=-\frac{12}{13}\left(\frac{2}{13}x_{3}+\frac{17}{13}\right)-\frac{6}{13}$$

Solve x_{3}=-\frac{12}{13}\left(\frac{2}{13}x_{3}+\frac{17}{13}\right)-\frac{6}{13} for x_{3}.

$$x_{3}=-\frac{282}{193}$$

Substitute -\frac{282}{193} for x_{3} in the equation x_{2}=\frac{2}{13}x_{3}+\frac{17}{13}.

$$x_{2}=\frac{2}{13}\left(-\frac{282}{193}\right)+\frac{17}{13}$$

Calculate x_{2} from x_{2}=\frac{2}{13}\left(-\frac{282}{193}\right)+\frac{17}{13}.

$$x_{2}=\frac{209}{193}$$

Substitute \frac{209}{193} for x_{2} and -\frac{282}{193} for x_{3} in the equation x_{1}=-x_{2}-x_{3}+1.

$$x_{1}=-\frac{209}{193}-\left(-\frac{282}{193}\right)+1$$

Calculate x_{1} from x_{1}=-\frac{209}{193}-\left(-\frac{282}{193}\right)+1.

$$x_{1}=\frac{266}{193}$$

The system is now solved.

$$x_{1}=\frac{266}{193} x_{2}=\frac{209}{193} x_{3}=-\frac{282}{193}$$