Solve {l}{3x-y+z=3}{-x+y+2z=0}{-5x+3y+2z=-2}

$$\left. \begin{array}{l}{ 3 x - y + z = 3 }\\{ - x + y + 2 z = 0 }\\{ - 5 x + 3 y + 2 z = - 2 }\end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=5$$

$$z=-1$$

Short Steps Using Substitution

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

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

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

$$-x+3x+z-3+2z=0 -5x+3\left(3x+z-3\right)+2z=-2$$

Solve these equations for x and z respectively.

$$x=\frac{3}{2}-\frac{3}{2}z z=-\frac{4}{5}x+\frac{7}{5}$$

Substitute \frac{3}{2}-\frac{3}{2}z for x in the equation z=-\frac{4}{5}x+\frac{7}{5}.

$$z=-\frac{4}{5}\left(\frac{3}{2}-\frac{3}{2}z\right)+\frac{7}{5}$$

Solve z=-\frac{4}{5}\left(\frac{3}{2}-\frac{3}{2}z\right)+\frac{7}{5} for z.

$$z=-1$$

Substitute -1 for z in the equation x=\frac{3}{2}-\frac{3}{2}z.

$$x=\frac{3}{2}-\frac{3}{2}\left(-1\right)$$

Calculate x from x=\frac{3}{2}-\frac{3}{2}\left(-1\right).

$$x=3$$

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

$$y=3\times 3-1-3$$

Calculate y from y=3\times 3-1-3.

$$y=5$$

The system is now solved.

$$x=3 y=5 z=-1$$