Solve {r}{x+y+z=22}{12x+18y+20z=792}{x/2+y/3+z/4=15}

$$\left.\begin{array} { r } { x + y + z = 22 } \\ { 12 x + 18 y + 20 z = 792 } \\ { \frac { x } { 2 } + \frac { y } { 3 } + \frac { z } { 4 } = 15 } \end{array} \right\}$$

Solve for x, y, z

$$x=-290$$

$$y=984$$

$$z=-672$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$x+y+z=22 12x+18y+20z=792 6x+4y+3z=180$$

Solve x+y+z=22 for x.

$$x=-y-z+22$$

Substitute -y-z+22 for x in the second and third equation.

$$12\left(-y-z+22\right)+18y+20z=792 6\left(-y-z+22\right)+4y+3z=180$$

Solve these equations for y and z respectively.

$$y=88-\frac{4}{3}z z=-16-\frac{2}{3}y$$

Substitute 88-\frac{4}{3}z for y in the equation z=-16-\frac{2}{3}y.

$$z=-16-\frac{2}{3}\left(88-\frac{4}{3}z\right)$$

Solve z=-16-\frac{2}{3}\left(88-\frac{4}{3}z\right) for z.

$$z=-672$$

Substitute -672 for z in the equation y=88-\frac{4}{3}z.

$$y=88-\frac{4}{3}\left(-672\right)$$

Calculate y from y=88-\frac{4}{3}\left(-672\right).

$$y=984$$

Substitute 984 for y and -672 for z in the equation x=-y-z+22.

$$x=-984-\left(-672\right)+22$$

Calculate x from x=-984-\left(-672\right)+22.

$$x=-290$$

The system is now solved.

$$x=-290 y=984 z=-672$$