Solve {l}{x-3y+4z=22}{2x+y+z=2}{-4x+3y-4z=-25}

$$\left\{ \begin{array}{l}{ x - 3 y + 4 z = 22 }\\{ 2 x + y + z = 2 }\\{ - 4 x + 3 y - 4 z = - 25 }\end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-3$$

$$z=3$$

Short Steps Using Substitution

Solve x-3y+4z=22 for x.

$$x=3y-4z+22$$

Substitute 3y-4z+22 for x in the second and third equation.

$$2\left(3y-4z+22\right)+y+z=2 -4\left(3y-4z+22\right)+3y-4z=-25$$

Solve these equations for y and z respectively.

$$y=-6+z z=\frac{21}{4}+\frac{3}{4}y$$

Substitute -6+z for y in the equation z=\frac{21}{4}+\frac{3}{4}y.

$$z=\frac{21}{4}+\frac{3}{4}\left(-6+z\right)$$

Solve z=\frac{21}{4}+\frac{3}{4}\left(-6+z\right) for z.

$$z=3$$

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

$$y=-6+3$$

Calculate y from y=-6+3.

$$y=-3$$

Substitute -3 for y and 3 for z in the equation x=3y-4z+22.

$$x=3\left(-3\right)-4\times 3+22$$

Calculate x from x=3\left(-3\right)-4\times 3+22.

$$x=1$$

The system is now solved.

$$x=1 y=-3 z=3$$