Solve {r}{x-2y+z=0}{3y+z=9}{2x+y=4}

$$\left. \begin{array} { r } { x - 2 y + z = 0 } \\ { 3 y + z = 9 } \\ { 2 x + y = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

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

$$x=2y-z$$

Substitute 2y-z for x in the equation 2x+y=4.

$$2\left(2y-z\right)+y=4$$

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

$$y=-\frac{1}{3}z+3 z=-2+\frac{5}{2}y$$

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

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

Solve z=-2+\frac{5}{2}\left(-\frac{1}{3}z+3\right) for z.

$$z=3$$

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

$$y=-\frac{1}{3}\times 3+3$$

Calculate y from y=-\frac{1}{3}\times 3+3.

$$y=2$$

Substitute 2 for y and 3 for z in the equation x=2y-z.

$$x=2\times 2-3$$

Calculate x from x=2\times 2-3.

$$x=1$$

The system is now solved.

$$x=1 y=2 z=3$$