Solve {l}{x+4y+2z=9}{4x-3y+z=-38}{-3x+y-4z=32}

$$\left. \begin{array}{l}{ x + 4 y + 2 z = 9 }\\{ 4 x - 3 y + z = - 38 }\\{ - 3 x + y - 4 z = 32 }\end{array} \right.$$

Solve for x, y, z

$$x=-5$$

$$y=5$$

$$z=-3$$

Short Steps Using Substitution

Solve x+4y+2z=9 for x.

$$x=-4y-2z+9$$

Substitute -4y-2z+9 for x in the second and third equation.

$$4\left(-4y-2z+9\right)-3y+z=-38 -3\left(-4y-2z+9\right)+y-4z=32$$

Solve these equations for y and z respectively.

$$y=-\frac{7}{19}z+\frac{74}{19} z=-\frac{13}{2}y+\frac{59}{2}$$

Substitute -\frac{7}{19}z+\frac{74}{19} for y in the equation z=-\frac{13}{2}y+\frac{59}{2}.

$$z=-\frac{13}{2}\left(-\frac{7}{19}z+\frac{74}{19}\right)+\frac{59}{2}$$

Solve z=-\frac{13}{2}\left(-\frac{7}{19}z+\frac{74}{19}\right)+\frac{59}{2} for z.

$$z=-3$$

Substitute -3 for z in the equation y=-\frac{7}{19}z+\frac{74}{19}.

$$y=-\frac{7}{19}\left(-3\right)+\frac{74}{19}$$

Calculate y from y=-\frac{7}{19}\left(-3\right)+\frac{74}{19}.

$$y=5$$

Substitute 5 for y and -3 for z in the equation x=-4y-2z+9.

$$x=-4\times 5-2\left(-3\right)+9$$

Calculate x from x=-4\times 5-2\left(-3\right)+9.

$$x=-5$$

The system is now solved.

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