Solve {c}{-25x+5y+5z=-25}{5x+7y-2z=30}{5x-2y+7z=-20}

$$\left. \begin{array} { c } { - 25 x + 5 y + 5 z = - 25 } \\ { 5 x + 7 y - 2 z = 30 } \\ { 5 x - 2 y + 7 z = - 20 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y = \frac{25}{9} = 2\frac{7}{9} \approx 2.777777778$$

$$z = -\frac{25}{9} = -2\frac{7}{9} \approx -2.777777778$$

Short Steps Using Substitution

Solve -25x+5y+5z=-25 for x.

$$x=1+\frac{1}{5}y+\frac{1}{5}z$$

Substitute 1+\frac{1}{5}y+\frac{1}{5}z for x in the second and third equation.

$$5\left(1+\frac{1}{5}y+\frac{1}{5}z\right)+7y-2z=30 5\left(1+\frac{1}{5}y+\frac{1}{5}z\right)-2y+7z=-20$$

Solve these equations for y and z respectively.

$$y=\frac{25}{8}+\frac{1}{8}z z=-\frac{25}{8}+\frac{1}{8}y$$

Substitute \frac{25}{8}+\frac{1}{8}z for y in the equation z=-\frac{25}{8}+\frac{1}{8}y.

$$z=-\frac{25}{8}+\frac{1}{8}\left(\frac{25}{8}+\frac{1}{8}z\right)$$

Solve z=-\frac{25}{8}+\frac{1}{8}\left(\frac{25}{8}+\frac{1}{8}z\right) for z.

$$z=-\frac{25}{9}$$

Substitute -\frac{25}{9} for z in the equation y=\frac{25}{8}+\frac{1}{8}z.

$$y=\frac{25}{8}+\frac{1}{8}\left(-\frac{25}{9}\right)$$

Calculate y from y=\frac{25}{8}+\frac{1}{8}\left(-\frac{25}{9}\right).

$$y=\frac{25}{9}$$

Substitute \frac{25}{9} for y and -\frac{25}{9} for z in the equation x=1+\frac{1}{5}y+\frac{1}{5}z.

$$x=1+\frac{1}{5}\times \frac{25}{9}+\frac{1}{5}\left(-\frac{25}{9}\right)$$

Calculate x from x=1+\frac{1}{5}\times \frac{25}{9}+\frac{1}{5}\left(-\frac{25}{9}\right).

$$x=1$$

The system is now solved.

$$x=1 y=\frac{25}{9} z=-\frac{25}{9}$$