Solve {l}{x+y+z=16}{x+2y+3z=30}{2x+y+4z=36}

$$\left. \begin{array} { l } { x + y + z = 16 } \\ { x + 2 y + 3 z = 30 } \\ { 2 x + y + 4 z = 36 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{13}{2} = 6\frac{1}{2} = 6.5$$

$$y=5$$

$$z = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Short Steps Using Substitution

Solve x+y+z=16 for x.

$$x=-y-z+16$$

Substitute -y-z+16 for x in the second and third equation.

$$-y-z+16+2y+3z=30 2\left(-y-z+16\right)+y+4z=36$$

Solve these equations for y and z respectively.

$$y=14-2z z=\frac{1}{2}y+2$$

Substitute 14-2z for y in the equation z=\frac{1}{2}y+2.

$$z=\frac{1}{2}\left(14-2z\right)+2$$

Solve z=\frac{1}{2}\left(14-2z\right)+2 for z.

$$z=\frac{9}{2}$$

Substitute \frac{9}{2} for z in the equation y=14-2z.

$$y=14-2\times \frac{9}{2}$$

Calculate y from y=14-2\times \frac{9}{2}.

$$y=5$$

Substitute 5 for y and \frac{9}{2} for z in the equation x=-y-z+16.

$$x=-5-\frac{9}{2}+16$$

Calculate x from x=-5-\frac{9}{2}+16.

$$x=\frac{13}{2}$$

The system is now solved.

$$x=\frac{13}{2} y=5 z=\frac{9}{2}$$