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

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

Solve for x, y, z

$$x=5$$

$$y=6$$

$$z=7$$

Short Steps Using Substitution

Reorder the equations.

$$3x-2y-z=-4 -4x+3y+2z=12 -5x-y+5z=4$$

Solve 3x-2y-z=-4 for z.

$$z=3x-2y+4$$

Substitute 3x-2y+4 for z in the second and third equation.

$$-4x+3y+2\left(3x-2y+4\right)=12 -5x-y+5\left(3x-2y+4\right)=4$$

Solve these equations for y and x respectively.

$$y=-4+2x x=\frac{11}{10}y-\frac{8}{5}$$

Substitute -4+2x for y in the equation x=\frac{11}{10}y-\frac{8}{5}.

$$x=\frac{11}{10}\left(-4+2x\right)-\frac{8}{5}$$

Solve x=\frac{11}{10}\left(-4+2x\right)-\frac{8}{5} for x.

$$x=5$$

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

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

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

$$y=6$$

Substitute 6 for y and 5 for x in the equation z=3x-2y+4.

$$z=3\times 5-2\times 6+4$$

Calculate z from z=3\times 5-2\times 6+4.

$$z=7$$

The system is now solved.

$$x=5 y=6 z=7$$