Solve {c}{x+2y+3z=0}{3x+4y+4z=0}{7x+10y+12z=0}

$$\left. \begin{array} { c } { x + 2 y + 3 z = 0 } \\ { 3 x + 4 y + 4 z = 0 } \\ { 7 x + 10 y + 12 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=0$$

$$z=0$$

Short Steps Using Substitution

Solve x+2y+3z=0 for x.

$$x=-2y-3z$$

Substitute -2y-3z for x in the second and third equation.

$$3\left(-2y-3z\right)+4y+4z=0 7\left(-2y-3z\right)+10y+12z=0$$

Solve these equations for y and z respectively.

$$y=-\frac{5}{2}z z=-\frac{4}{9}y$$

Substitute -\frac{5}{2}z for y in the equation z=-\frac{4}{9}y.

$$z=-\frac{4}{9}\left(-\frac{5}{2}\right)z$$

Solve z=-\frac{4}{9}\left(-\frac{5}{2}\right)z for z.

$$z=0$$

Substitute 0 for z in the equation y=-\frac{5}{2}z.

$$y=-\frac{5}{2}\times 0$$

Calculate y from y=-\frac{5}{2}\times 0.

$$y=0$$

Substitute 0 for y and 0 for z in the equation x=-2y-3z.

$$x=-2\times 0-3\times 0$$

Calculate x from x=-2\times 0-3\times 0.

$$x=0$$

The system is now solved.

$$x=0 y=0 z=0$$