Solve {l}{x+y-z=11}{-2x+3y-z=9}{x-2y+z=7}

$$\left\{ \begin{array} { l } { x + y - z = 11 } \\ { - 2 x + 3 y - z = 9 } \\ { x - 2 y + z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x=34$$

$$y=50$$

$$z=73$$

Short Steps Using Substitution

Solve x+y-z=11 for x.

$$x=-y+z+11$$

Substitute -y+z+11 for x in the second and third equation.

$$-2\left(-y+z+11\right)+3y-z=9 -y+z+11-2y+z=7$$

Solve these equations for y and z respectively.

$$y=\frac{3}{5}z+\frac{31}{5} z=\frac{3}{2}y-2$$

Substitute \frac{3}{5}z+\frac{31}{5} for y in the equation z=\frac{3}{2}y-2.

$$z=\frac{3}{2}\left(\frac{3}{5}z+\frac{31}{5}\right)-2$$

Solve z=\frac{3}{2}\left(\frac{3}{5}z+\frac{31}{5}\right)-2 for z.

$$z=73$$

Substitute 73 for z in the equation y=\frac{3}{5}z+\frac{31}{5}.

$$y=\frac{3}{5}\times 73+\frac{31}{5}$$

Calculate y from y=\frac{3}{5}\times 73+\frac{31}{5}.

$$y=50$$

Substitute 50 for y and 73 for z in the equation x=-y+z+11.

$$x=-50+73+11$$

Calculate x from x=-50+73+11.

$$x=34$$

The system is now solved.

$$x=34 y=50 z=73$$