Solve {c}{-2x+4y-2z=2}{3x-5y+z=-7}{2x-5z=-16}

$$\left\{ \begin{array} { c } { - 2 x + 4 y - 2 z = 2 } \\ { 3 x - 5 y + z = - 7 } \\ { 2 x - 5 z = - 16 } \end{array} \right.$$

Solve for x, y, z

$$x=-3$$

$$y=0$$

$$z=2$$

Short Steps Using Substitution

Reorder the equations.

$$3x-5y+z=-7 -2x+4y-2z=2 2x-5z=-16$$

Solve 3x-5y+z=-7 for z.

$$z=-3x+5y-7$$

Substitute -3x+5y-7 for z in the second and third equation.

$$-2x+4y-2\left(-3x+5y-7\right)=2 2x-5\left(-3x+5y-7\right)=-16$$

Solve these equations for y and x respectively.

$$y=2+\frac{2}{3}x x=\frac{25}{17}y-3$$

Substitute 2+\frac{2}{3}x for y in the equation x=\frac{25}{17}y-3.

$$x=\frac{25}{17}\left(2+\frac{2}{3}x\right)-3$$

Solve x=\frac{25}{17}\left(2+\frac{2}{3}x\right)-3 for x.

$$x=-3$$

Substitute -3 for x in the equation y=2+\frac{2}{3}x.

$$y=2+\frac{2}{3}\left(-3\right)$$

Calculate y from y=2+\frac{2}{3}\left(-3\right).

$$y=0$$

Substitute 0 for y and -3 for x in the equation z=-3x+5y-7.

$$z=-3\left(-3\right)+5\times 0-7$$

Calculate z from z=-3\left(-3\right)+5\times 0-7.

$$z=2$$

The system is now solved.

$$x=-3 y=0 z=2$$