Solve {l}{-3x-2y+2z=9}{x-3y+3z=19}{x-3y-2z=14}

$$\left. \begin{array} { l } { - 3 x - 2 y + 2 z = 9 } \\ { x - 3 y + 3 z = 19 } \\ { x - 3 y - 2 z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-5$$

$$z=1$$

Short Steps Using Substitution

Reorder the equations.

$$x-3y+3z=19 -3x-2y+2z=9 x-3y-2z=14$$

Solve x-3y+3z=19 for x.

$$x=3y-3z+19$$

Substitute 3y-3z+19 for x in the second and third equation.

$$-3\left(3y-3z+19\right)-2y+2z=9 3y-3z+19-3y-2z=14$$

Solve these equations for y and z respectively.

$$y=-6+z z=1$$

Substitute 1 for z in the equation y=-6+z.

$$y=-6+1$$

Calculate y from y=-6+1.

$$y=-5$$

Substitute -5 for y and 1 for z in the equation x=3y-3z+19.

$$x=3\left(-5\right)-3+19$$

Calculate x from x=3\left(-5\right)-3+19.

$$x=1$$

The system is now solved.

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