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

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

Solve for x, y, z

$$x = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

$$y = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$z = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

Short Steps Using Substitution

Reorder the equations.

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

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

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

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

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

Solve these equations for y and z respectively.

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

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

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

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

$$z=-\frac{9}{2}$$

Substitute -\frac{9}{2} for z in the equation y=-\frac{1}{7}z+\frac{13}{7}.

$$y=-\frac{1}{7}\left(-\frac{9}{2}\right)+\frac{13}{7}$$

Calculate y from y=-\frac{1}{7}\left(-\frac{9}{2}\right)+\frac{13}{7}.

$$y=\frac{5}{2}$$

Substitute \frac{5}{2} for y and -\frac{9}{2} for z in the equation x=2y-z-4.

$$x=2\times \frac{5}{2}-\left(-\frac{9}{2}\right)-4$$

Calculate x from x=2\times \frac{5}{2}-\left(-\frac{9}{2}\right)-4.

$$x=\frac{11}{2}$$

The system is now solved.

$$x=\frac{11}{2} y=\frac{5}{2} z=-\frac{9}{2}$$