Solve {l}{2x-3y+7z=31}{x-2y+4z=18}{x-y+2z=11}

$$\left. \begin{array} { l } { 2 x - 3 y + 7 z = 31 } \\ { x - 2 y + 4 z = 18 } \\ { x - y + 2 z = 11 } \end{array} \right.$$

Solve for x, y, z

$$x=4$$

$$y=-3$$

$$z=2$$

Short Steps Using Substitution

Reorder the equations.

$$x-2y+4z=18 2x-3y+7z=31 x-y+2z=11$$

Solve x-2y+4z=18 for x.

$$x=18+2y-4z$$

Substitute 18+2y-4z for x in the second and third equation.

$$2\left(18+2y-4z\right)-3y+7z=31 18+2y-4z-y+2z=11$$

Solve these equations for y and z respectively.

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

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

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

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

$$z=2$$

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

$$y=-5+2$$

Calculate y from y=-5+2.

$$y=-3$$

Substitute -3 for y and 2 for z in the equation x=18+2y-4z.

$$x=18+2\left(-3\right)-4\times 2$$

Calculate x from x=18+2\left(-3\right)-4\times 2.

$$x=4$$

The system is now solved.

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