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

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

Solve for x, y, z

$$x=-1$$

$$y=3$$

$$z=5$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for y and z respectively.

$$y=\frac{6}{7}+\frac{3}{7}z z=\frac{11}{5}y-\frac{8}{5}$$

Substitute \frac{6}{7}+\frac{3}{7}z for y in the equation z=\frac{11}{5}y-\frac{8}{5}.

$$z=\frac{11}{5}\left(\frac{6}{7}+\frac{3}{7}z\right)-\frac{8}{5}$$

Solve z=\frac{11}{5}\left(\frac{6}{7}+\frac{3}{7}z\right)-\frac{8}{5} for z.

$$z=5$$

Substitute 5 for z in the equation y=\frac{6}{7}+\frac{3}{7}z.

$$y=\frac{6}{7}+\frac{3}{7}\times 5$$

Calculate y from y=\frac{6}{7}+\frac{3}{7}\times 5.

$$y=3$$

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

$$x=-3\times 3+5+3$$

Calculate x from x=-3\times 3+5+3.

$$x=-1$$

The system is now solved.

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