Solve {c}{2x-y+z=3}{-3x+y+z=2}{5x+y-2z=1}

$$\left. \begin{array} { c } { 2 x - y + z = 3 } \\ { - 3 x + y + z = 2 } \\ { 5 x + y - 2 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

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

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

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

$$-3x+2x+z-3+z=2 5x+2x+z-3-2z=1$$

Solve these equations for x and z respectively.

$$x=2z-5 z=7x-4$$

Substitute 2z-5 for x in the equation z=7x-4.

$$z=7\left(2z-5\right)-4$$

Solve z=7\left(2z-5\right)-4 for z.

$$z=3$$

Substitute 3 for z in the equation x=2z-5.

$$x=2\times 3-5$$

Calculate x from x=2\times 3-5.

$$x=1$$

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

$$y=2\times 1+3-3$$

Calculate y from y=2\times 1+3-3.

$$y=2$$

The system is now solved.

$$x=1 y=2 z=3$$