Solve {l}{1/4x_{1}+1/5x_{2}+1/6x_{3}=9}{1/3x_{1}+frac{1}{4}x_{2}+frac{1}{5}x_{3}=8}{frac{1}{2}x_{1}+x_{2}+2x_{3}=8}

$$\left\{ \begin{array} { l } { \frac { 1 } { 4 } x _ { 1 } + \frac { 1 } { 5 } x _ { 2 } + \frac { 1 } { 6 } x _ { 3 } = 9 } \\ { \frac { 1 } { 3 } x _ { 1 } + \frac { 1 } { 4 } x _ { 2 } + \frac { 1 } { 5 } x _ { 3 } = 8 } \\ { \frac { 1 } { 2 } x _ { 1 } + x _ { 2 } + 2 x _ { 3 } = 8 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = -\frac{2952}{13} = -227\frac{1}{13} \approx -227.076923077$$

$$x_{2} = \frac{6200}{13} = 476\frac{12}{13} \approx 476.923076923$$

$$x_{3} = -\frac{2310}{13} = -177\frac{9}{13} \approx -177.692307692$$

Short Steps Using Substitution

Reorder the equations.

$$\frac{1}{2}x_{1}+x_{2}+2x_{3}=8 \frac{1}{3}x_{1}+\frac{1}{4}x_{2}+\frac{1}{5}x_{3}=8 \frac{1}{4}x_{1}+\frac{1}{5}x_{2}+\frac{1}{6}x_{3}=9$$

Solve \frac{1}{2}x_{1}+x_{2}+2x_{3}=8 for x_{2}.

$$x_{2}=8-\frac{1}{2}x_{1}-2x_{3}$$

Substitute 8-\frac{1}{2}x_{1}-2x_{3} for x_{2} in the second and third equation.

$$\frac{1}{3}x_{1}+\frac{1}{4}\left(8-\frac{1}{2}x_{1}-2x_{3}\right)+\frac{1}{5}x_{3}=8 \frac{1}{4}x_{1}+\frac{1}{5}\left(8-\frac{1}{2}x_{1}-2x_{3}\right)+\frac{1}{6}x_{3}=9$$

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

$$x_{1}=\frac{144}{5}+\frac{36}{25}x_{3} x_{3}=-\frac{222}{7}+\frac{9}{14}x_{1}$$

Substitute \frac{144}{5}+\frac{36}{25}x_{3} for x_{1} in the equation x_{3}=-\frac{222}{7}+\frac{9}{14}x_{1}.

$$x_{3}=-\frac{222}{7}+\frac{9}{14}\left(\frac{144}{5}+\frac{36}{25}x_{3}\right)$$

Solve x_{3}=-\frac{222}{7}+\frac{9}{14}\left(\frac{144}{5}+\frac{36}{25}x_{3}\right) for x_{3}.

$$x_{3}=-\frac{2310}{13}$$

Substitute -\frac{2310}{13} for x_{3} in the equation x_{1}=\frac{144}{5}+\frac{36}{25}x_{3}.

$$x_{1}=\frac{144}{5}+\frac{36}{25}\left(-\frac{2310}{13}\right)$$

Calculate x_{1} from x_{1}=\frac{144}{5}+\frac{36}{25}\left(-\frac{2310}{13}\right).

$$x_{1}=-\frac{2952}{13}$$

Substitute -\frac{2952}{13} for x_{1} and -\frac{2310}{13} for x_{3} in the equation x_{2}=8-\frac{1}{2}x_{1}-2x_{3}.

$$x_{2}=8-\frac{1}{2}\left(-\frac{2952}{13}\right)-2\left(-\frac{2310}{13}\right)$$

Calculate x_{2} from x_{2}=8-\frac{1}{2}\left(-\frac{2952}{13}\right)-2\left(-\frac{2310}{13}\right).

$$x_{2}=\frac{6200}{13}$$

The system is now solved.

$$x_{1}=-\frac{2952}{13} x_{2}=\frac{6200}{13} x_{3}=-\frac{2310}{13}$$