Solve {l}{2x+2y+z=3}{3x+2y-2z=-1}{5x-2y-6z=17}

$$\left. \begin{array} { l } { 2 x + 2 y + z = 3 } \\ { 3 x + 2 y - 2 z = - 1 } \\ { 5 x - 2 y - 6 z = 17 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=-5$$

$$z=3$$

Short Steps Using Substitution

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

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

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

$$3x+2y-2\left(-2x-2y+3\right)=-1 5x-2y-6\left(-2x-2y+3\right)=17$$

Solve these equations for y and x respectively.

$$y=-\frac{7}{6}x+\frac{5}{6} x=\frac{35}{17}-\frac{10}{17}y$$

Substitute -\frac{7}{6}x+\frac{5}{6} for y in the equation x=\frac{35}{17}-\frac{10}{17}y.

$$x=\frac{35}{17}-\frac{10}{17}\left(-\frac{7}{6}x+\frac{5}{6}\right)$$

Solve x=\frac{35}{17}-\frac{10}{17}\left(-\frac{7}{6}x+\frac{5}{6}\right) for x.

$$x=5$$

Substitute 5 for x in the equation y=-\frac{7}{6}x+\frac{5}{6}.

$$y=-\frac{7}{6}\times 5+\frac{5}{6}$$

Calculate y from y=-\frac{7}{6}\times 5+\frac{5}{6}.

$$y=-5$$

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

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

Calculate z from z=-2\times 5-2\left(-5\right)+3.

$$z=3$$

The system is now solved.

$$x=5 y=-5 z=3$$