Solve {l}{x+y+2=5}{2x-y+z=9}{-2y+3z=16}

$$\left. \begin{array} { l } { x + y + 2 = 5 } \\ { 2 x - y + z = 9 } \\ { - 2 y + 3 z = 16 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=1$$

$$z=6$$

Short Steps Using Substitution

Solve x+y+2=5 for x.

$$x=-y+3$$

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

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

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

$$y=-1+\frac{1}{3}z z=\frac{16}{3}+\frac{2}{3}y$$

Substitute -1+\frac{1}{3}z for y in the equation z=\frac{16}{3}+\frac{2}{3}y.

$$z=\frac{16}{3}+\frac{2}{3}\left(-1+\frac{1}{3}z\right)$$

Solve z=\frac{16}{3}+\frac{2}{3}\left(-1+\frac{1}{3}z\right) for z.

$$z=6$$

Substitute 6 for z in the equation y=-1+\frac{1}{3}z.

$$y=-1+\frac{1}{3}\times 6$$

Calculate y from y=-1+\frac{1}{3}\times 6.

$$y=1$$

Substitute 1 for y in the equation x=-y+3.

$$x=-1+3$$

Calculate x from x=-1+3.

$$x=2$$

The system is now solved.

$$x=2 y=1 z=6$$