Solve {c}{-2y+3z=4}{2x+3z=3}{x-6y+6z=9}

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

Solve for y, z, x

$$x=0$$

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

$$z=1$$

Short Steps Using Substitution

Reorder the equations.

$$x-6y+6z=9 2x+3z=3 -2y+3z=4$$

Solve x-6y+6z=9 for x.

$$x=9+6y-6z$$

Substitute 9+6y-6z for x in the equation 2x+3z=3.

$$2\left(9+6y-6z\right)+3z=3$$

Solve the second equation for z and the third equation for y.

$$z=\frac{5}{3}+\frac{4}{3}y y=\frac{3}{2}z-2$$

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

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

Solve y=\frac{3}{2}\left(\frac{5}{3}+\frac{4}{3}y\right)-2 for y.

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

Substitute -\frac{1}{2} for y in the equation z=\frac{5}{3}+\frac{4}{3}y.

$$z=\frac{5}{3}+\frac{4}{3}\left(-\frac{1}{2}\right)$$

Calculate z from z=\frac{5}{3}+\frac{4}{3}\left(-\frac{1}{2}\right).

$$z=1$$

Substitute 1 for z and -\frac{1}{2} for y in the equation x=9+6y-6z.

$$x=9+6\left(-\frac{1}{2}\right)-6$$

Calculate x from x=9+6\left(-\frac{1}{2}\right)-6.

$$x=0$$

The system is now solved.

$$y=-\frac{1}{2} z=1 x=0$$