Solve {r}{-9x+4y-6z=10}{17x-14y-2z=-16}{x-16y-2z=-56}

$$\left. \begin{array} { r } { - 9 x + 4 y - 6 z = 10 } \\ { 17 x - 14 y - 2 z = - 16 } \\ { x - 16 y - 2 z = - 56 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{431}{214} = 2\frac{3}{214} \approx 2.014018692$$

$$y = \frac{416}{107} = 3\frac{95}{107} \approx 3.887850467$$

$$z = -\frac{897}{428} = -2\frac{41}{428} \approx -2.095794393$$

Short Steps Using Substitution

Reorder the equations.

$$x-16y-2z=-56 17x-14y-2z=-16 -9x+4y-6z=10$$

Solve x-16y-2z=-56 for x.

$$x=-56+16y+2z$$

Substitute -56+16y+2z for x in the second and third equation.

$$17\left(-56+16y+2z\right)-14y-2z=-16 -9\left(-56+16y+2z\right)+4y-6z=10$$

Solve these equations for y and z respectively.

$$y=\frac{156}{43}-\frac{16}{129}z z=\frac{247}{12}-\frac{35}{6}y$$

Substitute \frac{156}{43}-\frac{16}{129}z for y in the equation z=\frac{247}{12}-\frac{35}{6}y.

$$z=\frac{247}{12}-\frac{35}{6}\left(\frac{156}{43}-\frac{16}{129}z\right)$$

Solve z=\frac{247}{12}-\frac{35}{6}\left(\frac{156}{43}-\frac{16}{129}z\right) for z.

$$z=-\frac{897}{428}$$

Substitute -\frac{897}{428} for z in the equation y=\frac{156}{43}-\frac{16}{129}z.

$$y=\frac{156}{43}-\frac{16}{129}\left(-\frac{897}{428}\right)$$

Calculate y from y=\frac{156}{43}-\frac{16}{129}\left(-\frac{897}{428}\right).

$$y=\frac{416}{107}$$

Substitute \frac{416}{107} for y and -\frac{897}{428} for z in the equation x=-56+16y+2z.

$$x=-56+16\times \frac{416}{107}+2\left(-\frac{897}{428}\right)$$

Calculate x from x=-56+16\times \frac{416}{107}+2\left(-\frac{897}{428}\right).

$$x=\frac{431}{214}$$

The system is now solved.

$$x=\frac{431}{214} y=\frac{416}{107} z=-\frac{897}{428}$$