Solve {l}{3x-y-2z=-15}{x+3y-z=2}{2x+y-5z=-3}

$$\left\{ \begin{array} { l } { 3 x - y - 2 z = - 15 } \\ { x + 3 y - z = 2 } \\ { 2 x + y - 5 z = - 3 } \end{array} \right.$$

Solve for x, y, z

$$x=-5$$

$$y=2$$

$$z=-1$$

Short Steps Using Substitution

Solve 3x-y-2z=-15 for y.

$$y=3x-2z+15$$

Substitute 3x-2z+15 for y in the second and third equation.

$$x+3\left(3x-2z+15\right)-z=2 2x+3x-2z+15-5z=-3$$

Solve these equations for x and z respectively.

$$x=\frac{7}{10}z-\frac{43}{10} z=\frac{5}{7}x+\frac{18}{7}$$

Substitute \frac{7}{10}z-\frac{43}{10} for x in the equation z=\frac{5}{7}x+\frac{18}{7}.

$$z=\frac{5}{7}\left(\frac{7}{10}z-\frac{43}{10}\right)+\frac{18}{7}$$

Solve z=\frac{5}{7}\left(\frac{7}{10}z-\frac{43}{10}\right)+\frac{18}{7} for z.

$$z=-1$$

Substitute -1 for z in the equation x=\frac{7}{10}z-\frac{43}{10}.

$$x=\frac{7}{10}\left(-1\right)-\frac{43}{10}$$

Calculate x from x=\frac{7}{10}\left(-1\right)-\frac{43}{10}.

$$x=-5$$

Substitute -5 for x and -1 for z in the equation y=3x-2z+15.

$$y=3\left(-5\right)-2\left(-1\right)+15$$

Calculate y from y=3\left(-5\right)-2\left(-1\right)+15.

$$y=2$$

The system is now solved.

$$x=-5 y=2 z=-1$$