Solve {l}{x+2y=3}{y-2z=-6}{-3x-5y+6z=-31}

$$\left\{ \begin{array}{l}{ x + 2 y = 3 }\\{ y - 2 z = - 6 }\\{ - 3 x - 5 y + 6 z = - 31 }\end{array} \right.$$

Solve for x, y, z

$$x=23$$

$$y=-10$$

$$z=-2$$

Short Steps Using Substitution

Solve x+2y=3 for x.

$$x=-2y+3$$

Substitute -2y+3 for x in the equation -3x-5y+6z=-31.

$$-3\left(-2y+3\right)-5y+6z=-31$$

Solve the second equation for y and the third equation for z.

$$y=-6+2z z=-\frac{1}{6}y-\frac{11}{3}$$

Substitute -6+2z for y in the equation z=-\frac{1}{6}y-\frac{11}{3}.

$$z=-\frac{1}{6}\left(-6+2z\right)-\frac{11}{3}$$

Solve z=-\frac{1}{6}\left(-6+2z\right)-\frac{11}{3} for z.

$$z=-2$$

Substitute -2 for z in the equation y=-6+2z.

$$y=-6+2\left(-2\right)$$

Calculate y from y=-6+2\left(-2\right).

$$y=-10$$

Substitute -10 for y in the equation x=-2y+3.

$$x=-2\left(-10\right)+3$$

Calculate x from x=-2\left(-10\right)+3.

$$x=23$$

The system is now solved.

$$x=23 y=-10 z=-2$$