Solve {l}{x+y+z=10}{x+3y+2z=17}{-x+2y+3z=8}

$$\left. \begin{array} { l } { x + y + z = 10 } \\ { x + 3 y + 2 z = 17 } \\ { - x + 2 y + 3 z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

Solve x+y+z=10 for x.

$$x=-y-z+10$$

Substitute -y-z+10 for x in the second and third equation.

$$-y-z+10+3y+2z=17 -\left(-y-z+10\right)+2y+3z=8$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{2}z+\frac{7}{2} z=\frac{9}{2}-\frac{3}{4}y$$

Substitute -\frac{1}{2}z+\frac{7}{2} for y in the equation z=\frac{9}{2}-\frac{3}{4}y.

$$z=\frac{9}{2}-\frac{3}{4}\left(-\frac{1}{2}z+\frac{7}{2}\right)$$

Solve z=\frac{9}{2}-\frac{3}{4}\left(-\frac{1}{2}z+\frac{7}{2}\right) for z.

$$z=3$$

Substitute 3 for z in the equation y=-\frac{1}{2}z+\frac{7}{2}.

$$y=-\frac{1}{2}\times 3+\frac{7}{2}$$

Calculate y from y=-\frac{1}{2}\times 3+\frac{7}{2}.

$$y=2$$

Substitute 2 for y and 3 for z in the equation x=-y-z+10.

$$x=-2-3+10$$

Calculate x from x=-2-3+10.

$$x=5$$

The system is now solved.

$$x=5 y=2 z=3$$