Solve {l}{2x+5y+2z=6}{5x-7y-3z=-7}{4x-9y-2z=-22}

$$\left. \begin{array} { l } { 2 x + 5 y + 2 z = 6 } \\ { 5 x - 7 y - 3 z = - 7 } \\ { 4 x - 9 y - 2 z = - 22 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=4$$

$$z=-7$$

Short Steps Using Substitution

Solve 2x+5y+2z=6 for x.

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

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

$$5\left(-\frac{5}{2}y-z+3\right)-7y-3z=-7 4\left(-\frac{5}{2}y-z+3\right)-9y-2z=-22$$

Solve these equations for y and z respectively.

$$y=\frac{44}{39}-\frac{16}{39}z z=-\frac{19}{6}y+\frac{17}{3}$$

Substitute \frac{44}{39}-\frac{16}{39}z for y in the equation z=-\frac{19}{6}y+\frac{17}{3}.

$$z=-\frac{19}{6}\left(\frac{44}{39}-\frac{16}{39}z\right)+\frac{17}{3}$$

Solve z=-\frac{19}{6}\left(\frac{44}{39}-\frac{16}{39}z\right)+\frac{17}{3} for z.

$$z=-7$$

Substitute -7 for z in the equation y=\frac{44}{39}-\frac{16}{39}z.

$$y=\frac{44}{39}-\frac{16}{39}\left(-7\right)$$

Calculate y from y=\frac{44}{39}-\frac{16}{39}\left(-7\right).

$$y=4$$

Substitute 4 for y and -7 for z in the equation x=-\frac{5}{2}y-z+3.

$$x=-\frac{5}{2}\times 4-\left(-7\right)+3$$

Calculate x from x=-\frac{5}{2}\times 4-\left(-7\right)+3.

$$x=0$$

The system is now solved.

$$x=0 y=4 z=-7$$