Solve {l}{106x+137y+86z=4865}{x+y+2z=47}{-2y+z=33}

$$\left. \begin{array} { l } { 106 x + 137 y + 86 z = 4865 } \\ { x + y + 2 z = 47 } \\ { - 2 y + z = 33 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{16006}{221} = 72\frac{94}{221} \approx 72.425339367$$

$$y = -\frac{4041}{221} = -18\frac{63}{221} \approx -18.285067873$$

$$z = -\frac{789}{221} = -3\frac{126}{221} \approx -3.570135747$$

Short Steps Using Substitution

Reorder the equations.

$$x+y+2z=47 106x+137y+86z=4865 -2y+z=33$$

Solve x+y+2z=47 for x.

$$x=-y-2z+47$$

Substitute -y-2z+47 for x in the equation 106x+137y+86z=4865.

$$106\left(-y-2z+47\right)+137y+86z=4865$$

Solve the second equation for y and the third equation for z.

$$y=-\frac{117}{31}+\frac{126}{31}z z=2y+33$$

Substitute -\frac{117}{31}+\frac{126}{31}z for y in the equation z=2y+33.

$$z=2\left(-\frac{117}{31}+\frac{126}{31}z\right)+33$$

Solve z=2\left(-\frac{117}{31}+\frac{126}{31}z\right)+33 for z.

$$z=-\frac{789}{221}$$

Substitute -\frac{789}{221} for z in the equation y=-\frac{117}{31}+\frac{126}{31}z.

$$y=-\frac{117}{31}+\frac{126}{31}\left(-\frac{789}{221}\right)$$

Calculate y from y=-\frac{117}{31}+\frac{126}{31}\left(-\frac{789}{221}\right).

$$y=-\frac{4041}{221}$$

Substitute -\frac{4041}{221} for y and -\frac{789}{221} for z in the equation x=-y-2z+47.

$$x=-\left(-\frac{4041}{221}\right)-2\left(-\frac{789}{221}\right)+47$$

Calculate x from x=-\left(-\frac{4041}{221}\right)-2\left(-\frac{789}{221}\right)+47.

$$x=\frac{16006}{221}$$

The system is now solved.

$$x=\frac{16006}{221} y=-\frac{4041}{221} z=-\frac{789}{221}$$