Solve {l}{x+y+2z=12}{2x+2y+z=18}{y+2x+z=15}

$$\left\{ \begin{array} { l } { x + y + 2 z = 12 } \\ { 2 x + 2 y + z = 18 } \\ { y + 2 x + z = 15 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=3$$

$$z=2$$

Short Steps Using Substitution

Solve x+y+2z=12 for x.

$$x=-y-2z+12$$

Substitute -y-2z+12 for x in the second and third equation.

$$2\left(-y-2z+12\right)+2y+z=18 y+2\left(-y-2z+12\right)+z=15$$

Solve these equations for z and y respectively.

$$z=2 y=-3z+9$$

Substitute 2 for z in the equation y=-3z+9.

$$y=-3\times 2+9$$

Calculate y from y=-3\times 2+9.

$$y=3$$

Substitute 2 for z and 3 for y in the equation x=-y-2z+12.

$$x=-3-2\times 2+12$$

Calculate x from x=-3-2\times 2+12.

$$x=5$$

The system is now solved.

$$x=5 y=3 z=2$$