Solve {l}{6x+9y-3z=0}{4x-7y+z=0}{3x-2y-3z=12}

$$\left. \begin{array} { l } { 6 x + 9 y - 3 z = 0 } \\ { 4 x - 7 y + z = 0 } \\ { 3 x - 2 y - 3 z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{8}{13}\approx -0.615384615$$

$$y=-\frac{12}{13}\approx -0.923076923$$

$$z=-4$$

Short Steps Using Substitution

Reorder the equations.

$$4x-7y+z=0 6x+9y-3z=0 3x-2y-3z=12$$

Solve 4x-7y+z=0 for z.

$$z=-4x+7y$$

Substitute -4x+7y for z in the second and third equation.

$$6x+9y-3\left(-4x+7y\right)=0 3x-2y-3\left(-4x+7y\right)=12$$

Solve these equations for y and x respectively.

$$y=\frac{3}{2}x x=\frac{23}{15}y+\frac{4}{5}$$

Substitute \frac{3}{2}x for y in the equation x=\frac{23}{15}y+\frac{4}{5}.

$$x=\frac{23}{15}\times \frac{3}{2}x+\frac{4}{5}$$

Solve x=\frac{23}{15}\times \frac{3}{2}x+\frac{4}{5} for x.

$$x=-\frac{8}{13}$$

Substitute -\frac{8}{13} for x in the equation y=\frac{3}{2}x.

$$y=\frac{3}{2}\left(-\frac{8}{13}\right)$$

Calculate y from y=\frac{3}{2}\left(-\frac{8}{13}\right).

$$y=-\frac{12}{13}$$

Substitute -\frac{12}{13} for y and -\frac{8}{13} for x in the equation z=-4x+7y.

$$z=-4\left(-\frac{8}{13}\right)+7\left(-\frac{12}{13}\right)$$

Calculate z from z=-4\left(-\frac{8}{13}\right)+7\left(-\frac{12}{13}\right).

$$z=-4$$

The system is now solved.

$$x=-\frac{8}{13} y=-\frac{12}{13} z=-4$$