Solve {l}{x+y+2z=0}{x-y-3z=5}{x-5y-9z=3}

$$\left. \begin{array} { l } { x + y + 2 z = 0 } \\ { x - y - 3 z = 5 } \\ { x - 5 y - 9 z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=5$$

$$z=-3$$

Short Steps Using Substitution

Solve x+y+2z=0 for x.

$$x=-y-2z$$

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

$$-y-2z-y-3z=5 -y-2z-5y-9z=3$$

Solve these equations for y and z respectively.

$$y=-\frac{5}{2}-\frac{5}{2}z z=-\frac{3}{11}-\frac{6}{11}y$$

Substitute -\frac{5}{2}-\frac{5}{2}z for y in the equation z=-\frac{3}{11}-\frac{6}{11}y.

$$z=-\frac{3}{11}-\frac{6}{11}\left(-\frac{5}{2}-\frac{5}{2}z\right)$$

Solve z=-\frac{3}{11}-\frac{6}{11}\left(-\frac{5}{2}-\frac{5}{2}z\right) for z.

$$z=-3$$

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

$$y=-\frac{5}{2}-\frac{5}{2}\left(-3\right)$$

Calculate y from y=-\frac{5}{2}-\frac{5}{2}\left(-3\right).

$$y=5$$

Substitute 5 for y and -3 for z in the equation x=-y-2z.

$$x=-5-2\left(-3\right)$$

Calculate x from x=-5-2\left(-3\right).

$$x=1$$

The system is now solved.

$$x=1 y=5 z=-3$$