Solve {l}{x+y+z=11}{2x-y+z=5}{3x+2y+z=24}

$$\left\{ \begin{array} { l } { x + y + z = 11 } \\ { 2 x - y + z = 5 } \\ { 3 x + 2 y + z = 24 } \end{array} \right.$$

Solve for x, y, z

$$x=4$$

$$y=5$$

$$z=2$$

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.

$$2\left(-y-z+11\right)-y+z=5 3\left(-y-z+11\right)+2y+z=24$$

Solve these equations for y and z respectively.

$$y=\frac{17}{3}-\frac{1}{3}z z=-\frac{1}{2}y+\frac{9}{2}$$

Substitute \frac{17}{3}-\frac{1}{3}z for y in the equation z=-\frac{1}{2}y+\frac{9}{2}.

$$z=-\frac{1}{2}\left(\frac{17}{3}-\frac{1}{3}z\right)+\frac{9}{2}$$

Solve z=-\frac{1}{2}\left(\frac{17}{3}-\frac{1}{3}z\right)+\frac{9}{2} for z.

$$z=2$$

Substitute 2 for z in the equation y=\frac{17}{3}-\frac{1}{3}z.

$$y=\frac{17}{3}-\frac{1}{3}\times 2$$

Calculate y from y=\frac{17}{3}-\frac{1}{3}\times 2.

$$y=5$$

Substitute 5 for y and 2 for z in the equation x=-y-z+11.

$$x=-5-2+11$$

Calculate x from x=-5-2+11.

$$x=4$$

The system is now solved.

$$x=4 y=5 z=2$$