Solve {l}{6x-2y+3z=0}{3x+6y-2z=0}{x+y+3=7}

$$\left\{ \begin{array} { l } { 6 x - 2 y + 3 z = 0 } \\ { 3 x + 6 y - 2 z = 0 } \\ { x + y + 3 = 7 } \end{array} \right.$$

Solve for x, y, z

$$x=-8$$

$$y=12$$

$$z=24$$

Short Steps Using Substitution

Reorder the equations.

$$x+y+3=7 3x+6y-2z=0 6x-2y+3z=0$$

Solve x+y+3=7 for x.

$$x=-y+4$$

Substitute -y+4 for x in the second and third equation.

$$3\left(-y+4\right)+6y-2z=0 6\left(-y+4\right)-2y+3z=0$$

Solve these equations for y and z respectively.

$$y=-4+\frac{2}{3}z z=-8+\frac{8}{3}y$$

Substitute -4+\frac{2}{3}z for y in the equation z=-8+\frac{8}{3}y.

$$z=-8+\frac{8}{3}\left(-4+\frac{2}{3}z\right)$$

Solve z=-8+\frac{8}{3}\left(-4+\frac{2}{3}z\right) for z.

$$z=24$$

Substitute 24 for z in the equation y=-4+\frac{2}{3}z.

$$y=-4+\frac{2}{3}\times 24$$

Calculate y from y=-4+\frac{2}{3}\times 24.

$$y=12$$

Substitute 12 for y in the equation x=-y+4.

$$x=-12+4$$

Calculate x from x=-12+4.

$$x=-8$$

The system is now solved.

$$x=-8 y=12 z=24$$