Solve {l}{x+y+z=6}{-3x-4y+4z=13}{2x-y-4z=-3}

$$\left. \begin{array}{l}{ x + y + z = 6 }\\{ - 3 x - 4 y + 4 z = 13 }\\{ 2 x - y - 4 z = - 3 }\end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=-3$$

$$z=4$$

Short Steps Using Substitution

Solve x+y+z=6 for x.

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

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

$$-3\left(-y-z+6\right)-4y+4z=13 2\left(-y-z+6\right)-y-4z=-3$$

Solve these equations for y and z respectively.

$$y=7z-31 z=\frac{5}{2}-\frac{1}{2}y$$

Substitute 7z-31 for y in the equation z=\frac{5}{2}-\frac{1}{2}y.

$$z=\frac{5}{2}-\frac{1}{2}\left(7z-31\right)$$

Solve z=\frac{5}{2}-\frac{1}{2}\left(7z-31\right) for z.

$$z=4$$

Substitute 4 for z in the equation y=7z-31.

$$y=7\times 4-31$$

Calculate y from y=7\times 4-31.

$$y=-3$$

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

$$x=-\left(-3\right)-4+6$$

Calculate x from x=-\left(-3\right)-4+6.

$$x=5$$

The system is now solved.

$$x=5 y=-3 z=4$$