Solve {l}{2x-2y+z=-5}{x+2y=3z+3}{z=2x+1}

$$\left. \begin{array} { l } { 2 x - 2 y + z = - 5 } \\ { x + 2 y = 3 z + 3 } \\ { z = 2 x + 1 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=3$$

$$z=1$$

Short Steps Using Substitution

Reorder the equations.

$$z=2x+1 x+2y=3z+3 2x-2y+z=-5$$

Substitute 2x+1 for z in the second and third equation.

$$x+2y=3\left(2x+1\right)+3 2x-2y+2x+1=-5$$

Solve these equations for y and x respectively.

$$y=\frac{5}{2}x+3 x=-\frac{3}{2}+\frac{1}{2}y$$

Substitute \frac{5}{2}x+3 for y in the equation x=-\frac{3}{2}+\frac{1}{2}y.

$$x=-\frac{3}{2}+\frac{1}{2}\left(\frac{5}{2}x+3\right)$$

Solve x=-\frac{3}{2}+\frac{1}{2}\left(\frac{5}{2}x+3\right) for x.

$$x=0$$

Substitute 0 for x in the equation y=\frac{5}{2}x+3.

$$y=\frac{5}{2}\times 0+3$$

Calculate y from y=\frac{5}{2}\times 0+3.

$$y=3$$

Substitute 0 for x in the equation z=2x+1.

$$z=2\times 0+1$$

Calculate z from z=2\times 0+1.

$$z=1$$

The system is now solved.

$$x=0 y=3 z=1$$