Solve {l}{2x+4y+6z=-12}{2x-3y-4z=15}{3x+4y+5z=-8}

$$\left\{ \begin{array} { l } { 2 x + 4 y + 6 z = - 12 } \\ { 2 x - 3 y - 4 z = 15 } \\ { 3 x + 4 y + 5 z = - 8 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-1$$

$$z=-2$$

Short Steps Using Substitution

Solve 2x+4y+6z=-12 for x.

$$x=-6-2y-3z$$

Substitute -6-2y-3z for x in the second and third equation.

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

Solve these equations for y and z respectively.

$$y=-\frac{27}{7}-\frac{10}{7}z z=-\frac{5}{2}-\frac{1}{2}y$$

Substitute -\frac{27}{7}-\frac{10}{7}z for y in the equation z=-\frac{5}{2}-\frac{1}{2}y.

$$z=-\frac{5}{2}-\frac{1}{2}\left(-\frac{27}{7}-\frac{10}{7}z\right)$$

Solve z=-\frac{5}{2}-\frac{1}{2}\left(-\frac{27}{7}-\frac{10}{7}z\right) for z.

$$z=-2$$

Substitute -2 for z in the equation y=-\frac{27}{7}-\frac{10}{7}z.

$$y=-\frac{27}{7}-\frac{10}{7}\left(-2\right)$$

Calculate y from y=-\frac{27}{7}-\frac{10}{7}\left(-2\right).

$$y=-1$$

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

$$x=-6-2\left(-1\right)-3\left(-2\right)$$

Calculate x from x=-6-2\left(-1\right)-3\left(-2\right).

$$x=2$$

The system is now solved.

$$x=2 y=-1 z=-2$$