Solve {r}{2x+y-4z=-1}{x+10z=5}{4x+y+6z=1}

$$\left. \begin{array} { r } { 2 x + y - 4 z = - 1 } \\ { x + 10 z = 5 } \\ { 4 x + y + 6 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=-3$$

$$y = \frac{41}{5} = 8\frac{1}{5} = 8.2$$

$$z=\frac{4}{5}=0.8$$

Short Steps Using Substitution

Solve 2x+y-4z=-1 for y.

$$y=-2x+4z-1$$

Substitute -2x+4z-1 for y in the equation 4x+y+6z=1.

$$4x-2x+4z-1+6z=1$$

Solve the second equation for x and the third equation for z.

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

Substitute 5-10z for x in the equation z=\frac{1}{5}-\frac{1}{5}x.

$$z=\frac{1}{5}-\frac{1}{5}\left(5-10z\right)$$

Solve z=\frac{1}{5}-\frac{1}{5}\left(5-10z\right) for z.

$$z=\frac{4}{5}$$

Substitute \frac{4}{5} for z in the equation x=5-10z.

$$x=5-10\times \frac{4}{5}$$

Calculate x from x=5-10\times \frac{4}{5}.

$$x=-3$$

Substitute -3 for x and \frac{4}{5} for z in the equation y=-2x+4z-1.

$$y=-2\left(-3\right)+4\times \frac{4}{5}-1$$

Calculate y from y=-2\left(-3\right)+4\times \frac{4}{5}-1.

$$y=\frac{41}{5}$$

The system is now solved.

$$x=-3 y=\frac{41}{5} z=\frac{4}{5}$$