Solve {l}{x+y+z=0}{3x+2y-2z=-29}{-3x-y+4z=46}

$$\left. \begin{array}{l}{ x + y + z = 0 }\\{ 3 x + 2 y - 2 z = - 29 }\\{ - 3 x - y + 4 z = 46 }\end{array} \right.$$

Solve for x, y, z

$$x=-13$$

$$y=9$$

$$z=4$$

Short Steps Using Substitution

Solve x+y+z=0 for x.

$$x=-y-z$$

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

$$3\left(-y-z\right)+2y-2z=-29 -3\left(-y-z\right)-y+4z=46$$

Solve these equations for y and z respectively.

$$y=-5z+29 z=\frac{46}{7}-\frac{2}{7}y$$

Substitute -5z+29 for y in the equation z=\frac{46}{7}-\frac{2}{7}y.

$$z=\frac{46}{7}-\frac{2}{7}\left(-5z+29\right)$$

Solve z=\frac{46}{7}-\frac{2}{7}\left(-5z+29\right) for z.

$$z=4$$

Substitute 4 for z in the equation y=-5z+29.

$$y=-5\times 4+29$$

Calculate y from y=-5\times 4+29.

$$y=9$$

Substitute 9 for y and 4 for z in the equation x=-y-z.

$$x=-9-4$$

Calculate x from x=-9-4.

$$x=-13$$

The system is now solved.

$$x=-13 y=9 z=4$$