Solve {l}{x+y+z/4=0}{x/2+2y+z/9=0}{x/3+frac{3y+z}{16}=frac{1}{24}}

$$\left\{ \begin{array} { l } { x + \frac { y + z } { 4 } = 0 } \\ { \frac { x } { 2 } + \frac { 2 y + z } { 9 } = 0 } \\ { \frac { x } { 3 } + \frac { 3 y + z } { 16 } = \frac { 1 } { 24 } } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-1$$

$$z=-7$$

Short Steps Using Substitution

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

$$4x+y+z=0 9x+4y+2z=0 16x+9y+3z=2$$

Solve 4x+y+z=0 for y.

$$y=-4x-z$$

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

$$9x+4\left(-4x-z\right)+2z=0 16x+9\left(-4x-z\right)+3z=2$$

Solve these equations for x and z respectively.

$$x=-\frac{2}{7}z z=-\frac{1}{3}-\frac{10}{3}x$$

Substitute -\frac{2}{7}z for x in the equation z=-\frac{1}{3}-\frac{10}{3}x.

$$z=-\frac{1}{3}-\frac{10}{3}\left(-\frac{2}{7}\right)z$$

Solve z=-\frac{1}{3}-\frac{10}{3}\left(-\frac{2}{7}\right)z for z.

$$z=-7$$

Substitute -7 for z in the equation x=-\frac{2}{7}z.

$$x=-\frac{2}{7}\left(-7\right)$$

Calculate x from x=-\frac{2}{7}\left(-7\right).

$$x=2$$

Substitute 2 for x and -7 for z in the equation y=-4x-z.

$$y=-4\times 2-\left(-7\right)$$

Calculate y from y=-4\times 2-\left(-7\right).

$$y=-1$$

The system is now solved.

$$x=2 y=-1 z=-7$$