Solve {l}{x+y+z=0}{2x-y+z=-15}{x+2y-z=40}

$$\left\{ \begin{array} { l } { x + y + z = 0 } \\ { 2 x - y + z = - 15 } \\ { x + 2 y - z = 40 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=10$$

$$z=-15$$

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.

$$2\left(-y-z\right)-y+z=-15 -y-z+2y-z=40$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{3}z+5 z=-20+\frac{1}{2}y$$

Substitute -\frac{1}{3}z+5 for y in the equation z=-20+\frac{1}{2}y.

$$z=-20+\frac{1}{2}\left(-\frac{1}{3}z+5\right)$$

Solve z=-20+\frac{1}{2}\left(-\frac{1}{3}z+5\right) for z.

$$z=-15$$

Substitute -15 for z in the equation y=-\frac{1}{3}z+5.

$$y=-\frac{1}{3}\left(-15\right)+5$$

Calculate y from y=-\frac{1}{3}\left(-15\right)+5.

$$y=10$$

Substitute 10 for y and -15 for z in the equation x=-y-z.

$$x=-10-\left(-15\right)$$

Calculate x from x=-10-\left(-15\right).

$$x=5$$

The system is now solved.

$$x=5 y=10 z=-15$$