Solve {r}{2x+y-2z=-1}{3x-3y=5+z}{-2x+x+3z=5}

$$\left\{ \begin{array} { r } { 2 x + y - 2 z = - 1 } \\ { 3 x - 3 y = 5 + z } \\ { - 2 x + x + 3 z = 5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{41}{20} = 2\frac{1}{20} = 2.05$$

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

$$z = \frac{47}{20} = 2\frac{7}{20} = 2.35$$

Short Steps Using Substitution

Solve 2x+y-2z=-1 for y.

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

Substitute -2x+2z-1 for y in the equation 3x-3y=5+z.

$$3x-3\left(-2x+2z-1\right)=5+z$$

Solve the second equation for x and the third equation for z.

$$x=\frac{7}{9}z+\frac{2}{9} z=\frac{5}{3}+\frac{1}{3}x$$

Substitute \frac{7}{9}z+\frac{2}{9} for x in the equation z=\frac{5}{3}+\frac{1}{3}x.

$$z=\frac{5}{3}+\frac{1}{3}\left(\frac{7}{9}z+\frac{2}{9}\right)$$

Solve z=\frac{5}{3}+\frac{1}{3}\left(\frac{7}{9}z+\frac{2}{9}\right) for z.

$$z=\frac{47}{20}$$

Substitute \frac{47}{20} for z in the equation x=\frac{7}{9}z+\frac{2}{9}.

$$x=\frac{7}{9}\times \frac{47}{20}+\frac{2}{9}$$

Calculate x from x=\frac{7}{9}\times \frac{47}{20}+\frac{2}{9}.

$$x=\frac{41}{20}$$

Substitute \frac{41}{20} for x and \frac{47}{20} for z in the equation y=-2x+2z-1.

$$y=-2\times \frac{41}{20}+2\times \frac{47}{20}-1$$

Calculate y from y=-2\times \frac{41}{20}+2\times \frac{47}{20}-1.

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

The system is now solved.

$$x=\frac{41}{20} y=-\frac{2}{5} z=\frac{47}{20}$$