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

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

Solve for x, y, z

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

$$y=\frac{118}{221}\approx 0.533936652$$

$$z=\frac{43}{221}\approx 0.194570136$$

Short Steps Using Substitution

Reorder the equations.

$$6x+y+2z=0 5x+3y+6z=2 x+7y-3z=3$$

Solve 6x+y+2z=0 for y.

$$y=-6x-2z$$

Substitute -6x-2z for y in the second and third equation.

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

Solve these equations for x and z respectively.

$$x=-\frac{2}{13} z=-\frac{41}{17}x-\frac{3}{17}$$

Substitute -\frac{2}{13} for x in the equation z=-\frac{41}{17}x-\frac{3}{17}.

$$z=-\frac{41}{17}\left(-\frac{2}{13}\right)-\frac{3}{17}$$

Calculate z from z=-\frac{41}{17}\left(-\frac{2}{13}\right)-\frac{3}{17}.

$$z=\frac{43}{221}$$

Substitute -\frac{2}{13} for x and \frac{43}{221} for z in the equation y=-6x-2z.

$$y=-6\left(-\frac{2}{13}\right)-2\times \frac{43}{221}$$

Calculate y from y=-6\left(-\frac{2}{13}\right)-2\times \frac{43}{221}.

$$y=\frac{118}{221}$$

The system is now solved.

$$x=-\frac{2}{13} y=\frac{118}{221} z=\frac{43}{221}$$