Solve {c}{5x-y+2z=7}{-2x+6y+9z=0}{-7x+5y-3z=-7}

$$\left. \begin{array} { c } { 5 x - y + 2 z = 7 } \\ { - 2 x + 6 y + 9 z = 0 } \\ { - 7 x + 5 y - 3 z = - 7 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{21}{13} = 1\frac{8}{13} \approx 1.615384615$$

$$y=\frac{10}{13}\approx 0.769230769$$

$$z=-\frac{2}{13}\approx -0.153846154$$

Short Steps Using Substitution

Solve 5x-y+2z=7 for y.

$$y=5x+2z-7$$

Substitute 5x+2z-7 for y in the second and third equation.

$$-2x+6\left(5x+2z-7\right)+9z=0 -7x+5\left(5x+2z-7\right)-3z=-7$$

Solve these equations for x and z respectively.

$$x=\frac{3}{2}-\frac{3}{4}z z=4-\frac{18}{7}x$$

Substitute \frac{3}{2}-\frac{3}{4}z for x in the equation z=4-\frac{18}{7}x.

$$z=4-\frac{18}{7}\left(\frac{3}{2}-\frac{3}{4}z\right)$$

Solve z=4-\frac{18}{7}\left(\frac{3}{2}-\frac{3}{4}z\right) for z.

$$z=-\frac{2}{13}$$

Substitute -\frac{2}{13} for z in the equation x=\frac{3}{2}-\frac{3}{4}z.

$$x=\frac{3}{2}-\frac{3}{4}\left(-\frac{2}{13}\right)$$

Calculate x from x=\frac{3}{2}-\frac{3}{4}\left(-\frac{2}{13}\right).

$$x=\frac{21}{13}$$

Substitute \frac{21}{13} for x and -\frac{2}{13} for z in the equation y=5x+2z-7.

$$y=5\times \frac{21}{13}+2\left(-\frac{2}{13}\right)-7$$

Calculate y from y=5\times \frac{21}{13}+2\left(-\frac{2}{13}\right)-7.

$$y=\frac{10}{13}$$

The system is now solved.

$$x=\frac{21}{13} y=\frac{10}{13} z=-\frac{2}{13}$$