Solve x+y+z=5;2x-y+z=9;x-2y+3z=16

$$x + y + z = 5 ; 2 x - y + z = 9 ; x - 2 y + 3 z = 16$$

Solve for x, y, z

$$x=2$$

$$y=-1$$

$$z=4$$

Short Steps Using Substitution

Solve x+y+z=5 for x.

$$x=-y-z+5$$

Substitute -y-z+5 for x in the second and third equation.

$$2\left(-y-z+5\right)-y+z=9 -y-z+5-2y+3z=16$$

Solve these equations for y and z respectively.

$$y=\frac{1}{3}-\frac{1}{3}z z=\frac{3}{2}y+\frac{11}{2}$$

Substitute \frac{1}{3}-\frac{1}{3}z for y in the equation z=\frac{3}{2}y+\frac{11}{2}.

$$z=\frac{3}{2}\left(\frac{1}{3}-\frac{1}{3}z\right)+\frac{11}{2}$$

Solve z=\frac{3}{2}\left(\frac{1}{3}-\frac{1}{3}z\right)+\frac{11}{2} for z.

$$z=4$$

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

$$y=\frac{1}{3}-\frac{1}{3}\times 4$$

Calculate y from y=\frac{1}{3}-\frac{1}{3}\times 4.

$$y=-1$$

Substitute -1 for y and 4 for z in the equation x=-y-z+5.

$$x=-\left(-1\right)-4+5$$

Calculate x from x=-\left(-1\right)-4+5.

$$x=2$$

The system is now solved.

$$x=2 y=-1 z=4$$