Solve {l}{3x-4y-6z=-16}{4x-y-z=5}{x-3y-2z=-2}

$$\left. \begin{array} { l } { 3 x - 4 y - 6 z = - 16 } \\ { 4 x - y - z = 5 } \\ { x - 3 y - 2 z = - 2 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-2$$

$$z=5$$

Short Steps Using Substitution

Reorder the equations.

$$4x-y-z=5 3x-4y-6z=-16 x-3y-2z=-2$$

Solve 4x-y-z=5 for y.

$$y=4x-z-5$$

Substitute 4x-z-5 for y in the second and third equation.

$$3x-4\left(4x-z-5\right)-6z=-16 x-3\left(4x-z-5\right)-2z=-2$$

Solve these equations for x and z respectively.

$$x=\frac{36}{13}-\frac{2}{13}z z=11x-17$$

Substitute \frac{36}{13}-\frac{2}{13}z for x in the equation z=11x-17.

$$z=11\left(\frac{36}{13}-\frac{2}{13}z\right)-17$$

Solve z=11\left(\frac{36}{13}-\frac{2}{13}z\right)-17 for z.

$$z=5$$

Substitute 5 for z in the equation x=\frac{36}{13}-\frac{2}{13}z.

$$x=\frac{36}{13}-\frac{2}{13}\times 5$$

Calculate x from x=\frac{36}{13}-\frac{2}{13}\times 5.

$$x=2$$

Substitute 2 for x and 5 for z in the equation y=4x-z-5.

$$y=4\times 2-5-5$$

Calculate y from y=4\times 2-5-5.

$$y=-2$$

The system is now solved.

$$x=2 y=-2 z=5$$