Solve {r}{3x+4y-6z=-3}{x+y+z=0}{3x+y+3z=1}

$$\left. \begin{array} { r } { 3 x + 4 y - 6 z = - 3 } \\ { x + y + z = 0 } \\ { 3 x + y + 3 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{2}{9}\approx 0.222222222$$

$$y=-\frac{1}{2}=-0.5$$

$$z=\frac{5}{18}\approx 0.277777778$$

Short Steps Using Substitution

Reorder the equations.

$$x+y+z=0 3x+4y-6z=-3 3x+y+3z=1$$

Solve x+y+z=0 for x.

$$x=-y-z$$

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

$$3\left(-y-z\right)+4y-6z=-3 3\left(-y-z\right)+y+3z=1$$

Solve these equations for z and y respectively.

$$z=\frac{1}{9}y+\frac{1}{3} y=-\frac{1}{2}$$

Substitute -\frac{1}{2} for y in the equation z=\frac{1}{9}y+\frac{1}{3}.

$$z=\frac{1}{9}\left(-\frac{1}{2}\right)+\frac{1}{3}$$

Calculate z from z=\frac{1}{9}\left(-\frac{1}{2}\right)+\frac{1}{3}.

$$z=\frac{5}{18}$$

Substitute \frac{5}{18} for z and -\frac{1}{2} for y in the equation x=-y-z.

$$x=-\left(-\frac{1}{2}\right)-\frac{5}{18}$$

Calculate x from x=-\left(-\frac{1}{2}\right)-\frac{5}{18}.

$$x=\frac{2}{9}$$

The system is now solved.

$$x=\frac{2}{9} y=-\frac{1}{2} z=\frac{5}{18}$$