Solve {l}{x+2z=6}{2x+2y-4z=-12}{2x-3y+5z=15}

$$\left. \begin{array} { l } { x + 2 z = 6 } \\ { 2 x + 2 y - 4 z = - 12 } \\ { 2 x - 3 y + 5 z = 15 } \end{array} \right.$$

Solve for x, z, y

$$x=0$$

$$y=0$$

$$z=3$$

Short Steps Using Substitution

Solve x+2z=6 for x.

$$x=6-2z$$

Substitute 6-2z for x in the second and third equation.

$$2\left(6-2z\right)+2y-4z=-12 2\left(6-2z\right)-3y+5z=15$$

Solve these equations for z and y respectively.

$$z=3+\frac{1}{4}y y=-1+\frac{1}{3}z$$

Substitute 3+\frac{1}{4}y for z in the equation y=-1+\frac{1}{3}z.

$$y=-1+\frac{1}{3}\left(3+\frac{1}{4}y\right)$$

Solve y=-1+\frac{1}{3}\left(3+\frac{1}{4}y\right) for y.

$$y=0$$

Substitute 0 for y in the equation z=3+\frac{1}{4}y.

$$z=3+\frac{1}{4}\times 0$$

Calculate z from z=3+\frac{1}{4}\times 0.

$$z=3$$

Substitute 3 for z in the equation x=6-2z.

$$x=6-2\times 3$$

Calculate x from x=6-2\times 3.

$$x=0$$

The system is now solved.

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