Solve {l}{-x+3y+4z=0}{x+y-2z=-8}{-2x+4y+3z=-4}

$$\left. \begin{array}{l}{ - x + 3 y + 4 z = 0 }\\{ x + y - 2 z = - 8 }\\{ - 2 x + 4 y + 3 z = - 4 }\end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-3$$

$$z=2$$

Short Steps Using Substitution

Solve -x+3y+4z=0 for x.

$$x=3y+4z$$

Substitute 3y+4z for x in the second and third equation.

$$3y+4z+y-2z=-8 -2\left(3y+4z\right)+4y+3z=-4$$

Solve these equations for y and z respectively.

$$y=-2-\frac{1}{2}z z=\frac{4}{5}-\frac{2}{5}y$$

Substitute -2-\frac{1}{2}z for y in the equation z=\frac{4}{5}-\frac{2}{5}y.

$$z=\frac{4}{5}-\frac{2}{5}\left(-2-\frac{1}{2}z\right)$$

Solve z=\frac{4}{5}-\frac{2}{5}\left(-2-\frac{1}{2}z\right) for z.

$$z=2$$

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

$$y=-2-\frac{1}{2}\times 2$$

Calculate y from y=-2-\frac{1}{2}\times 2.

$$y=-3$$

Substitute -3 for y and 2 for z in the equation x=3y+4z.

$$x=3\left(-3\right)+4\times 2$$

Calculate x from x=3\left(-3\right)+4\times 2.

$$x=-1$$

The system is now solved.

$$x=-1 y=-3 z=2$$