Solve {l}{7x_{1}+9x_{2}+14x_{3}=8}{3x_{1}+8x_{2}+13x_{3}=79}{5x_{1}+3x_{2}+6x_{3}=25}

$$\left\{ \begin{array} { l } { 7 x _ { 1 } + 9 x _ { 2 } + 14 x _ { 3 } = 8 } \\ { 3 x _ { 1 } + 8 x _ { 2 } + 13 x _ { 3 } = 79 } \\ { 5 x _ { 1 } + 3 x _ { 2 } + 6 x _ { 3 } = 25 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = -\frac{751}{52} = -14\frac{23}{52} \approx -14.442307692$$

$$x_{2} = -\frac{3061}{52} = -58\frac{45}{52} \approx -58.865384615$$

$$x_{3} = \frac{2373}{52} = 45\frac{33}{52} \approx 45.634615385$$

Short Steps Using Substitution

Solve 7x_{1}+9x_{2}+14x_{3}=8 for x_{1}.

$$x_{1}=-\frac{9}{7}x_{2}-2x_{3}+\frac{8}{7}$$

Substitute -\frac{9}{7}x_{2}-2x_{3}+\frac{8}{7} for x_{1} in the second and third equation.

$$3\left(-\frac{9}{7}x_{2}-2x_{3}+\frac{8}{7}\right)+8x_{2}+13x_{3}=79 5\left(-\frac{9}{7}x_{2}-2x_{3}+\frac{8}{7}\right)+3x_{2}+6x_{3}=25$$

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

$$x_{2}=\frac{529}{29}-\frac{49}{29}x_{3} x_{3}=-\frac{135}{28}-\frac{6}{7}x_{2}$$

Substitute \frac{529}{29}-\frac{49}{29}x_{3} for x_{2} in the equation x_{3}=-\frac{135}{28}-\frac{6}{7}x_{2}.

$$x_{3}=-\frac{135}{28}-\frac{6}{7}\left(\frac{529}{29}-\frac{49}{29}x_{3}\right)$$

Solve x_{3}=-\frac{135}{28}-\frac{6}{7}\left(\frac{529}{29}-\frac{49}{29}x_{3}\right) for x_{3}.

$$x_{3}=\frac{2373}{52}$$

Substitute \frac{2373}{52} for x_{3} in the equation x_{2}=\frac{529}{29}-\frac{49}{29}x_{3}.

$$x_{2}=\frac{529}{29}-\frac{49}{29}\times \frac{2373}{52}$$

Calculate x_{2} from x_{2}=\frac{529}{29}-\frac{49}{29}\times \frac{2373}{52}.

$$x_{2}=-\frac{3061}{52}$$

Substitute -\frac{3061}{52} for x_{2} and \frac{2373}{52} for x_{3} in the equation x_{1}=-\frac{9}{7}x_{2}-2x_{3}+\frac{8}{7}.

$$x_{1}=-\frac{9}{7}\left(-\frac{3061}{52}\right)-2\times \frac{2373}{52}+\frac{8}{7}$$

Calculate x_{1} from x_{1}=-\frac{9}{7}\left(-\frac{3061}{52}\right)-2\times \frac{2373}{52}+\frac{8}{7}.

$$x_{1}=-\frac{751}{52}$$

The system is now solved.

$$x_{1}=-\frac{751}{52} x_{2}=-\frac{3061}{52} x_{3}=\frac{2373}{52}$$