Solve {l}{5x-2y-4z=3}{3x+3y+2z=-3}{-2x+5y+3z=3}

$$\left. \begin{array} { l } { 5 x - 2 y - 4 z = 3 } \\ { 3 x + 3 y + 2 z = - 3 } \\ { - 2 x + 5 y + 3 z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=2$$

$$z=-3$$

Short Steps Using Substitution

Solve 5x-2y-4z=3 for x.

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

Substitute \frac{2}{5}y+\frac{4}{5}z+\frac{3}{5} for x in the second and third equation.

$$3\left(\frac{2}{5}y+\frac{4}{5}z+\frac{3}{5}\right)+3y+2z=-3 -2\left(\frac{2}{5}y+\frac{4}{5}z+\frac{3}{5}\right)+5y+3z=3$$

Solve these equations for y and z respectively.

$$y=-\frac{8}{7}-\frac{22}{21}z z=3-3y$$

Substitute -\frac{8}{7}-\frac{22}{21}z for y in the equation z=3-3y.

$$z=3-3\left(-\frac{8}{7}-\frac{22}{21}z\right)$$

Solve z=3-3\left(-\frac{8}{7}-\frac{22}{21}z\right) for z.

$$z=-3$$

Substitute -3 for z in the equation y=-\frac{8}{7}-\frac{22}{21}z.

$$y=-\frac{8}{7}-\frac{22}{21}\left(-3\right)$$

Calculate y from y=-\frac{8}{7}-\frac{22}{21}\left(-3\right).

$$y=2$$

Substitute 2 for y and -3 for z in the equation x=\frac{2}{5}y+\frac{4}{5}z+\frac{3}{5}.

$$x=\frac{2}{5}\times 2+\frac{4}{5}\left(-3\right)+\frac{3}{5}$$

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

$$x=-1$$

The system is now solved.

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