Solve {l}{x+y+z=6}{2x+3y+z=11θ}{-3x+y+z=2(3)}

$$\left. \begin{array} { l } { x + y + z = 6 } \\ { 2 x + 3 y + z = 11 \theta } \\ { - 3 x + y + z = 2 ( 3 ) } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=\frac{11\theta }{2}-3$$

$$z=-\frac{11\theta }{2}+9$$

Short Steps Using Substitution

Solve x+y+z=6 for x.

$$x=-y-z+6$$

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

$$2\left(-y-z+6\right)+3y+z=11\theta -3\left(-y-z+6\right)+y+z=2\times 3$$

Solve these equations for y and z respectively.

$$y=z-12+11\theta z=6-y$$

Substitute z-12+11\theta for y in the equation z=6-y.

$$z=6-\left(z-12+11\theta \right)$$

Solve z=6-\left(z-12+11\theta \right) for z.

$$z=9-\frac{11}{2}\theta$$

Substitute 9-\frac{11}{2}\theta for z in the equation y=z-12+11\theta .

$$y=9-\frac{11}{2}\theta -12+11\theta$$

Calculate y from y=9-\frac{11}{2}\theta -12+11\theta .

$$y=-3+\frac{11}{2}\theta$$

Substitute -3+\frac{11}{2}\theta for y and 9-\frac{11}{2}\theta for z in the equation x=-y-z+6.

$$x=-\left(-3+\frac{11}{2}\theta \right)-\left(9-\frac{11}{2}\theta \right)+6$$

Calculate x from x=-\left(-3+\frac{11}{2}\theta \right)-\left(9-\frac{11}{2}\theta \right)+6.

$$x=0$$

The system is now solved.

$$x=0 y=-3+\frac{11}{2}\theta z=9-\frac{11}{2}\theta$$