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

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

Solve for x, y, z

$$x = \frac{13}{10} = 1\frac{3}{10} = 1.3$$

$$y=-\frac{2}{5}=-0.4$$

$$z=\frac{1}{10}=0.1$$

Short Steps Using Substitution

Reorder the equations.

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

Solve 2x+y-2z=2 for y.

$$y=2-2x+2z$$

Substitute 2-2x+2z for y in the second and third equation.

$$3x-2\left(2-2x+2z\right)+3z=5 2x+2\left(2-2x+2z\right)+2z=2$$

Solve these equations for x and z respectively.

$$x=\frac{9}{7}+\frac{1}{7}z z=-\frac{1}{3}+\frac{1}{3}x$$

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

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

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

$$z=\frac{1}{10}$$

Substitute \frac{1}{10} for z in the equation x=\frac{9}{7}+\frac{1}{7}z.

$$x=\frac{9}{7}+\frac{1}{7}\times \frac{1}{10}$$

Calculate x from x=\frac{9}{7}+\frac{1}{7}\times \frac{1}{10}.

$$x=\frac{13}{10}$$

Substitute \frac{13}{10} for x and \frac{1}{10} for z in the equation y=2-2x+2z.

$$y=2-2\times \frac{13}{10}+2\times \frac{1}{10}$$

Calculate y from y=2-2\times \frac{13}{10}+2\times \frac{1}{10}.

$$y=-\frac{2}{5}$$

The system is now solved.

$$x=\frac{13}{10} y=-\frac{2}{5} z=\frac{1}{10}$$