Solve {c}{6x-3y=21}{x+6y+2z=12}{3x+6y-z=-17}

$$\left. \begin{array} { c } { 6 x - 3 y = 21 } \\ { x + 6 y + 2 z = 12 } \\ { 3 x + 6 y - z = - 17 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{104}{43} = 2\frac{18}{43} \approx 2.418604651$$

$$y = -\frac{93}{43} = -2\frac{7}{43} \approx -2.162790698$$

$$z = \frac{485}{43} = 11\frac{12}{43} \approx 11.279069767$$

Short Steps Using Substitution

Reorder the equations.

$$x+6y+2z=12 6x-3y=21 3x+6y-z=-17$$

Solve x+6y+2z=12 for x.

$$x=12-6y-2z$$

Substitute 12-6y-2z for x in the second and third equation.

$$6\left(12-6y-2z\right)-3y=21 3\left(12-6y-2z\right)+6y-z=-17$$

Solve these equations for y and z respectively.

$$y=\frac{17}{13}-\frac{4}{13}z z=\frac{53}{7}-\frac{12}{7}y$$

Substitute \frac{17}{13}-\frac{4}{13}z for y in the equation z=\frac{53}{7}-\frac{12}{7}y.

$$z=\frac{53}{7}-\frac{12}{7}\left(\frac{17}{13}-\frac{4}{13}z\right)$$

Solve z=\frac{53}{7}-\frac{12}{7}\left(\frac{17}{13}-\frac{4}{13}z\right) for z.

$$z=\frac{485}{43}$$

Substitute \frac{485}{43} for z in the equation y=\frac{17}{13}-\frac{4}{13}z.

$$y=\frac{17}{13}-\frac{4}{13}\times \frac{485}{43}$$

Calculate y from y=\frac{17}{13}-\frac{4}{13}\times \frac{485}{43}.

$$y=-\frac{93}{43}$$

Substitute -\frac{93}{43} for y and \frac{485}{43} for z in the equation x=12-6y-2z.

$$x=12-6\left(-\frac{93}{43}\right)-2\times \frac{485}{43}$$

Calculate x from x=12-6\left(-\frac{93}{43}\right)-2\times \frac{485}{43}.

$$x=\frac{104}{43}$$

The system is now solved.

$$x=\frac{104}{43} y=-\frac{93}{43} z=\frac{485}{43}$$