Solve {l}{-3x+4y+3z=-16}{-2x-6y+4z=28}{x+y-3z=-8}

$$\left. \begin{array} { l } { - 3 x + 4 y + 3 z = - 16 } \\ { - 2 x - 6 y + 4 z = 28 } \\ { x + y - 3 z = - 8 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-4$$

$$z=2$$

Short Steps Using Substitution

Reorder the equations.

$$x+y-3z=-8 -2x-6y+4z=28 -3x+4y+3z=-16$$

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

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

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

$$-2\left(-y+3z-8\right)-6y+4z=28 -3\left(-y+3z-8\right)+4y+3z=-16$$

Solve these equations for y and z respectively.

$$y=-3-\frac{1}{2}z z=\frac{7}{6}y+\frac{20}{3}$$

Substitute -3-\frac{1}{2}z for y in the equation z=\frac{7}{6}y+\frac{20}{3}.

$$z=\frac{7}{6}\left(-3-\frac{1}{2}z\right)+\frac{20}{3}$$

Solve z=\frac{7}{6}\left(-3-\frac{1}{2}z\right)+\frac{20}{3} for z.

$$z=2$$

Substitute 2 for z in the equation y=-3-\frac{1}{2}z.

$$y=-3-\frac{1}{2}\times 2$$

Calculate y from y=-3-\frac{1}{2}\times 2.

$$y=-4$$

Substitute -4 for y and 2 for z in the equation x=-y+3z-8.

$$x=-\left(-4\right)+3\times 2-8$$

Calculate x from x=-\left(-4\right)+3\times 2-8.

$$x=2$$

The system is now solved.

$$x=2 y=-4 z=2$$