Solve {c}{x-y+3z=8}{2x-3y+z=-1}{3x+2y+(lambda-3)=-2}

$$\left.\begin{array} { c } { x - y + 3 z = 8 } \\ { 2 x - 3 y + z = - 1 } \\ { 3 x + 2 y + ( \lambda - 3 ) = - 2 } \end{array} \right\}$$

Solve for x, y, z

$$x=\frac{-4\lambda -7}{17}$$

$$y=-\frac{5\lambda }{34}+\frac{19}{17}$$

$$z=\frac{\lambda }{34}+\frac{54}{17}$$

Short Steps Using Substitution

Solve x-y+3z=8 for x.

$$x=y-3z+8$$

Substitute y-3z+8 for x in the second and third equation.

$$2\left(y-3z+8\right)-3y+z=-1 3\left(y-3z+8\right)+2y+\lambda -3=-2$$

Solve these equations for y and z respectively.

$$y=-5z+17 z=\frac{23}{9}+\frac{5}{9}y+\frac{1}{9}\lambda$$

Substitute -5z+17 for y in the equation z=\frac{23}{9}+\frac{5}{9}y+\frac{1}{9}\lambda .

$$z=\frac{23}{9}+\frac{5}{9}\left(-5z+17\right)+\frac{1}{9}\lambda$$

Solve z=\frac{23}{9}+\frac{5}{9}\left(-5z+17\right)+\frac{1}{9}\lambda for z.

$$z=\frac{54}{17}+\frac{1}{34}\lambda$$

Substitute \frac{54}{17}+\frac{1}{34}\lambda for z in the equation y=-5z+17.

$$y=-5\left(\frac{54}{17}+\frac{1}{34}\lambda \right)+17$$

Calculate y from y=-5\left(\frac{54}{17}+\frac{1}{34}\lambda \right)+17.

$$y=\frac{19}{17}-\frac{5}{34}\lambda$$

Substitute \frac{19}{17}-\frac{5}{34}\lambda for y and \frac{54}{17}+\frac{1}{34}\lambda for z in the equation x=y-3z+8.

$$x=\frac{19}{17}-\frac{5}{34}\lambda -3\left(\frac{54}{17}+\frac{1}{34}\lambda \right)+8$$

Calculate x from x=\frac{19}{17}-\frac{5}{34}\lambda -3\left(\frac{54}{17}+\frac{1}{34}\lambda \right)+8.

$$x=-\frac{7}{17}-\frac{4}{17}\lambda$$

The system is now solved.

$$x=-\frac{7}{17}-\frac{4}{17}\lambda y=\frac{19}{17}-\frac{5}{34}\lambda z=\frac{54}{17}+\frac{1}{34}\lambda$$