Solve {c}{x+3y-z=15}{2x+y+z=7}{x-y-2z=0}

$$\left. \begin{array} { c } { x + 3 y - z = 15 } \\ { 2 x + y + z = 7 } \\ { x - y - 2 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=4$$

$$z=-1$$

Short Steps Using Substitution

Solve x+3y-z=15 for x.

$$x=-3y+z+15$$

Substitute -3y+z+15 for x in the second and third equation.

$$2\left(-3y+z+15\right)+y+z=7 -3y+z+15-y-2z=0$$

Solve these equations for y and z respectively.

$$y=\frac{3}{5}z+\frac{23}{5} z=-4y+15$$

Substitute \frac{3}{5}z+\frac{23}{5} for y in the equation z=-4y+15.

$$z=-4\left(\frac{3}{5}z+\frac{23}{5}\right)+15$$

Solve z=-4\left(\frac{3}{5}z+\frac{23}{5}\right)+15 for z.

$$z=-1$$

Substitute -1 for z in the equation y=\frac{3}{5}z+\frac{23}{5}.

$$y=\frac{3}{5}\left(-1\right)+\frac{23}{5}$$

Calculate y from y=\frac{3}{5}\left(-1\right)+\frac{23}{5}.

$$y=4$$

Substitute 4 for y and -1 for z in the equation x=-3y+z+15.

$$x=-3\times 4-1+15$$

Calculate x from x=-3\times 4-1+15.

$$x=2$$

The system is now solved.

$$x=2 y=4 z=-1$$