Solve x+y-2z=7,2x-3y=0,x-2y-3z=3

$$x + y - 2 z = 7,2 x - 3 y = 0 , x - 2 y - 3 z = 3$$

Solve for x, y, z

$$x = \frac{45}{17} = 2\frac{11}{17} \approx 2.647058824$$

$$y = \frac{30}{17} = 1\frac{13}{17} \approx 1.764705882$$

$$z = -\frac{22}{17} = -1\frac{5}{17} \approx -1.294117647$$

Short Steps Using Substitution

Solve x+y-2z=7 for x.

$$x=-y+2z+7$$

Substitute -y+2z+7 for x in the second and third equation.

$$2\left(-y+2z+7\right)-3y=0 -y+2z+7-2y-3z=3$$

Solve these equations for y and z respectively.

$$y=\frac{14}{5}+\frac{4}{5}z z=-3y+4$$

Substitute \frac{14}{5}+\frac{4}{5}z for y in the equation z=-3y+4.

$$z=-3\left(\frac{14}{5}+\frac{4}{5}z\right)+4$$

Solve z=-3\left(\frac{14}{5}+\frac{4}{5}z\right)+4 for z.

$$z=-\frac{22}{17}$$

Substitute -\frac{22}{17} for z in the equation y=\frac{14}{5}+\frac{4}{5}z.

$$y=\frac{14}{5}+\frac{4}{5}\left(-\frac{22}{17}\right)$$

Calculate y from y=\frac{14}{5}+\frac{4}{5}\left(-\frac{22}{17}\right).

$$y=\frac{30}{17}$$

Substitute \frac{30}{17} for y and -\frac{22}{17} for z in the equation x=-y+2z+7.

$$x=-\frac{30}{17}+2\left(-\frac{22}{17}\right)+7$$

Calculate x from x=-\frac{30}{17}+2\left(-\frac{22}{17}\right)+7.

$$x=\frac{45}{17}$$

The system is now solved.

$$x=\frac{45}{17} y=\frac{30}{17} z=-\frac{22}{17}$$