Solve {l}{7x+6y=14}{6x+7z=18}{7z+14x=6}

$$\left. \begin{array} { l } { 7 x + 6 y = 14 } \\ { 6 x + 7 z = 18 } \\ { 7 z + 14 x = 6 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$y = \frac{49}{12} = 4\frac{1}{12} \approx 4.083333333$$

$$z = \frac{27}{7} = 3\frac{6}{7} \approx 3.857142857$$

Short Steps Using Substitution

Solve 7x+6y=14 for x.

$$x=2-\frac{6}{7}y$$

Substitute 2-\frac{6}{7}y for x in the second and third equation.

$$6\left(2-\frac{6}{7}y\right)+7z=18 7z+14\left(2-\frac{6}{7}y\right)=6$$

Solve these equations for y and z respectively.

$$y=-\frac{7}{6}+\frac{49}{36}z z=-\frac{22}{7}+\frac{12}{7}y$$

Substitute -\frac{7}{6}+\frac{49}{36}z for y in the equation z=-\frac{22}{7}+\frac{12}{7}y.

$$z=-\frac{22}{7}+\frac{12}{7}\left(-\frac{7}{6}+\frac{49}{36}z\right)$$

Solve z=-\frac{22}{7}+\frac{12}{7}\left(-\frac{7}{6}+\frac{49}{36}z\right) for z.

$$z=\frac{27}{7}$$

Substitute \frac{27}{7} for z in the equation y=-\frac{7}{6}+\frac{49}{36}z.

$$y=-\frac{7}{6}+\frac{49}{36}\times \frac{27}{7}$$

Calculate y from y=-\frac{7}{6}+\frac{49}{36}\times \frac{27}{7}.

$$y=\frac{49}{12}$$

Substitute \frac{49}{12} for y in the equation x=2-\frac{6}{7}y.

$$x=2-\frac{6}{7}\times \frac{49}{12}$$

Calculate x from x=2-\frac{6}{7}\times \frac{49}{12}.

$$x=-\frac{3}{2}$$

The system is now solved.

$$x=-\frac{3}{2} y=\frac{49}{12} z=\frac{27}{7}$$