Solve {l}{x+y+z=6}{x+2y+3z=14}{x+3y+9z=-34}

$$\left. \begin{array} { l } { x + y + z = 6 } \\ { x + 2 y + 3 z = 14 } \\ { x + 3 y + 9 z = - 34 } \end{array} \right.$$

Solve for x, y, z

$$x=-16$$

$$y=36$$

$$z=-14$$

Short Steps Using Substitution

Solve x+y+z=6 for x.

$$x=-y-z+6$$

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

$$-y-z+6+2y+3z=14 -y-z+6+3y+9z=-34$$

Solve these equations for y and z respectively.

$$y=8-2z z=-5-\frac{1}{4}y$$

Substitute 8-2z for y in the equation z=-5-\frac{1}{4}y.

$$z=-5-\frac{1}{4}\left(8-2z\right)$$

Solve z=-5-\frac{1}{4}\left(8-2z\right) for z.

$$z=-14$$

Substitute -14 for z in the equation y=8-2z.

$$y=8-2\left(-14\right)$$

Calculate y from y=8-2\left(-14\right).

$$y=36$$

Substitute 36 for y and -14 for z in the equation x=-y-z+6.

$$x=-36-\left(-14\right)+6$$

Calculate x from x=-36-\left(-14\right)+6.

$$x=-16$$

The system is now solved.

$$x=-16 y=36 z=-14$$