Solve 2x+3y+3z=5,x-2y+z=-4,3x-y-2z=3

$$2 x + 3 y + 3 z = 5 , x - 2 y + z = - 4,3 x - y - 2 z = 3$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=-1$$

Short Steps Using Substitution

Reorder the equations.

$$x-2y+z=-4 2x+3y+3z=5 3x-y-2z=3$$

Solve x-2y+z=-4 for x.

$$x=2y-z-4$$

Substitute 2y-z-4 for x in the second and third equation.

$$2\left(2y-z-4\right)+3y+3z=5 3\left(2y-z-4\right)-y-2z=3$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{7}z+\frac{13}{7} z=-3+y$$

Substitute -\frac{1}{7}z+\frac{13}{7} for y in the equation z=-3+y.

$$z=-3-\frac{1}{7}z+\frac{13}{7}$$

Solve z=-3-\frac{1}{7}z+\frac{13}{7} for z.

$$z=-1$$

Substitute -1 for z in the equation y=-\frac{1}{7}z+\frac{13}{7}.

$$y=-\frac{1}{7}\left(-1\right)+\frac{13}{7}$$

Calculate y from y=-\frac{1}{7}\left(-1\right)+\frac{13}{7}.

$$y=2$$

Substitute 2 for y and -1 for z in the equation x=2y-z-4.

$$x=2\times 2-\left(-1\right)-4$$

Calculate x from x=2\times 2-\left(-1\right)-4.

$$x=1$$

The system is now solved.

$$x=1 y=2 z=-1$$