Solve {r}{4x+2y+2z=18}{4x+2y-3z=-7}{6x-3y-5z=-1}

$$\left. \begin{array} { r } { 4 x + 2 y + 2 z = 18 } \\ { 4 x + 2 y - 3 z = - 7 } \\ { 6 x - 3 y - 5 z = - 1 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-2$$

$$z=5$$

Short Steps Using Substitution

Solve 4x+2y+2z=18 for x.

$$x=\frac{9}{2}-\frac{1}{2}y-\frac{1}{2}z$$

Substitute \frac{9}{2}-\frac{1}{2}y-\frac{1}{2}z for x in the second and third equation.

$$4\left(\frac{9}{2}-\frac{1}{2}y-\frac{1}{2}z\right)+2y-3z=-7 6\left(\frac{9}{2}-\frac{1}{2}y-\frac{1}{2}z\right)-3y-5z=-1$$

Solve these equations for z and y respectively.

$$z=5 y=\frac{14}{3}-\frac{4}{3}z$$

Substitute 5 for z in the equation y=\frac{14}{3}-\frac{4}{3}z.

$$y=\frac{14}{3}-\frac{4}{3}\times 5$$

Calculate y from y=\frac{14}{3}-\frac{4}{3}\times 5.

$$y=-2$$

Substitute 5 for z and -2 for y in the equation x=\frac{9}{2}-\frac{1}{2}y-\frac{1}{2}z.

$$x=\frac{9}{2}-\frac{1}{2}\left(-2\right)-\frac{1}{2}\times 5$$

Calculate x from x=\frac{9}{2}-\frac{1}{2}\left(-2\right)-\frac{1}{2}\times 5.

$$x=3$$

The system is now solved.

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