Solve {l}{5x+y+z=0}{3x+y-4z=1}{x+y+z=-2}

$$\left. \begin{array} { l } { 5 x + y + z = 0 } \\ { 3 x + y - 4 z = 1 } \\ { x + y + z = - 2 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{1}{2}=0.5$$

$$y = -\frac{21}{10} = -2\frac{1}{10} = -2.1$$

$$z=-\frac{2}{5}=-0.4$$

Short Steps Using Substitution

Solve 5x+y+z=0 for y.

$$y=-5x-z$$

Substitute -5x-z for y in the second and third equation.

$$3x-5x-z-4z=1 x-5x-z+z=-2$$

Solve these equations for z and x respectively.

$$z=-\frac{2}{5}x-\frac{1}{5} x=\frac{1}{2}$$

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

$$z=-\frac{2}{5}\times \frac{1}{2}-\frac{1}{5}$$

Calculate z from z=-\frac{2}{5}\times \frac{1}{2}-\frac{1}{5}.

$$z=-\frac{2}{5}$$

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

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

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

$$y=-\frac{21}{10}$$

The system is now solved.

$$x=\frac{1}{2} y=-\frac{21}{10} z=-\frac{2}{5}$$