Solve {l}{3x-y+z=9}{2x+y-2z=5}{-x+4y-z=0}

$$\left. \begin{array}{l}{ 3 x - y + z = 9 }\\{ 2 x + y - 2 z = 5 }\\{ - x + 4 y - z = 0 }\end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=1$$

$$z=1$$

Short Steps Using Substitution

Solve 3x-y+z=9 for y.

$$y=3x+z-9$$

Substitute 3x+z-9 for y in the second and third equation.

$$2x+3x+z-9-2z=5 -x+4\left(3x+z-9\right)-z=0$$

Solve these equations for x and z respectively.

$$x=\frac{14}{5}+\frac{1}{5}z z=-\frac{11}{3}x+12$$

Substitute \frac{14}{5}+\frac{1}{5}z for x in the equation z=-\frac{11}{3}x+12.

$$z=-\frac{11}{3}\left(\frac{14}{5}+\frac{1}{5}z\right)+12$$

Solve z=-\frac{11}{3}\left(\frac{14}{5}+\frac{1}{5}z\right)+12 for z.

$$z=1$$

Substitute 1 for z in the equation x=\frac{14}{5}+\frac{1}{5}z.

$$x=\frac{14}{5}+\frac{1}{5}\times 1$$

Calculate x from x=\frac{14}{5}+\frac{1}{5}\times 1.

$$x=3$$

Substitute 3 for x and 1 for z in the equation y=3x+z-9.

$$y=3\times 3+1-9$$

Calculate y from y=3\times 3+1-9.

$$y=1$$

The system is now solved.

$$x=3 y=1 z=1$$