Solve {l}{4x-y+6z=27}{-4x-2y+3z=21}{4x-6y+2z=12}

$$\left. \begin{array} { l } { 4 x - y + 6 z = 27 } \\ { - 4 x - 2 y + 3 z = 21 } \\ { 4 x - 6 y + 2 z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-1$$

$$z=5$$

Short Steps Using Substitution

Solve 4x-y+6z=27 for y.

$$y=4x+6z-27$$

Substitute 4x+6z-27 for y in the second and third equation.

$$-4x-2\left(4x+6z-27\right)+3z=21 4x-6\left(4x+6z-27\right)+2z=12$$

Solve these equations for x and z respectively.

$$x=\frac{11}{4}-\frac{3}{4}z z=\frac{75}{17}-\frac{10}{17}x$$

Substitute \frac{11}{4}-\frac{3}{4}z for x in the equation z=\frac{75}{17}-\frac{10}{17}x.

$$z=\frac{75}{17}-\frac{10}{17}\left(\frac{11}{4}-\frac{3}{4}z\right)$$

Solve z=\frac{75}{17}-\frac{10}{17}\left(\frac{11}{4}-\frac{3}{4}z\right) for z.

$$z=5$$

Substitute 5 for z in the equation x=\frac{11}{4}-\frac{3}{4}z.

$$x=\frac{11}{4}-\frac{3}{4}\times 5$$

Calculate x from x=\frac{11}{4}-\frac{3}{4}\times 5.

$$x=-1$$

Substitute -1 for x and 5 for z in the equation y=4x+6z-27.

$$y=4\left(-1\right)+6\times 5-27$$

Calculate y from y=4\left(-1\right)+6\times 5-27.

$$y=-1$$

The system is now solved.

$$x=-1 y=-1 z=5$$