Solve {l}{2x-6y-5z=30}{4x+6y-3z=-22}{-8x+6y+3z=2}{(x,y,z)=0}

$$\left. \begin{array} { l } { 2 x - 6 y - 5 z = 30 } \\ { 4 x + 6 y - 3 z = - 22 } \\ { - 8 x + 6 y + 3 z = 2 } \\ { ( x , y , z ) = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=-4$$

$$y=-3$$

$$z=-4$$

Short Steps Using Substitution

Solve 2x-6y-5z=30 for x.

$$x=3y+\frac{5}{2}z+15$$

Substitute 3y+\frac{5}{2}z+15 for x in the second and third equation.

$$4\left(3y+\frac{5}{2}z+15\right)+6y-3z=-22 -8\left(3y+\frac{5}{2}z+15\right)+6y+3z=2$$

Solve these equations for y and z respectively.

$$y=-\frac{7}{18}z-\frac{41}{9} z=-\frac{122}{17}-\frac{18}{17}y$$

Substitute -\frac{7}{18}z-\frac{41}{9} for y in the equation z=-\frac{122}{17}-\frac{18}{17}y.

$$z=-\frac{122}{17}-\frac{18}{17}\left(-\frac{7}{18}z-\frac{41}{9}\right)$$

Solve z=-\frac{122}{17}-\frac{18}{17}\left(-\frac{7}{18}z-\frac{41}{9}\right) for z.

$$z=-4$$

Substitute -4 for z in the equation y=-\frac{7}{18}z-\frac{41}{9}.

$$y=-\frac{7}{18}\left(-4\right)-\frac{41}{9}$$

Calculate y from y=-\frac{7}{18}\left(-4\right)-\frac{41}{9}.

$$y=-3$$

Substitute -3 for y and -4 for z in the equation x=3y+\frac{5}{2}z+15.

$$x=3\left(-3\right)+\frac{5}{2}\left(-4\right)+15$$

Calculate x from x=3\left(-3\right)+\frac{5}{2}\left(-4\right)+15.

$$x=-4$$

The system is now solved.

$$x=-4 y=-3 z=-4$$