Solve {l}{3x+6y-z=12}{2y-6z=-14}{x+y+2z=9}

$$\left. \begin{array} { l } { 3 x + 6 y - z = 12 } \\ { 2 y - 6 z = - 14 } \\ { x + y + 2 z = 9 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

Solve 3x+6y-z=12 for z.

$$z=-12+3x+6y$$

Substitute -12+3x+6y for z in the second and third equation.

$$2y-6\left(-12+3x+6y\right)=-14 x+y+2\left(-12+3x+6y\right)=9$$

Solve these equations for y and x respectively.

$$y=\frac{43}{17}-\frac{9}{17}x x=-\frac{13}{7}y+\frac{33}{7}$$

Substitute \frac{43}{17}-\frac{9}{17}x for y in the equation x=-\frac{13}{7}y+\frac{33}{7}.

$$x=-\frac{13}{7}\left(\frac{43}{17}-\frac{9}{17}x\right)+\frac{33}{7}$$

Solve x=-\frac{13}{7}\left(\frac{43}{17}-\frac{9}{17}x\right)+\frac{33}{7} for x.

$$x=1$$

Substitute 1 for x in the equation y=\frac{43}{17}-\frac{9}{17}x.

$$y=\frac{43}{17}-\frac{9}{17}$$

Calculate y from y=\frac{43}{17}-\frac{9}{17}.

$$y=2$$

Substitute 2 for y and 1 for x in the equation z=-12+3x+6y.

$$z=-12+3\times 1+6\times 2$$

Calculate z from z=-12+3\times 1+6\times 2.

$$z=3$$

The system is now solved.

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