Solve {l}{2x+y+6z=3}{x-y+4z=1}{3x+2y-2z=2}

$$\left\{ \begin{array}{l}{ 2 x + y + 6 z = 3 }\\{ x - y + 4 z = 1 }\\{ 3 x + 2 y - 2 z = 2 }\end{array} \right.$$

Solve for x, y, z

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

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

$$z=\frac{1}{4}=0.25$$

Short Steps Using Substitution

Solve 2x+y+6z=3 for y.

$$y=-2x-6z+3$$

Substitute -2x-6z+3 for y in the second and third equation.

$$x-\left(-2x-6z+3\right)+4z=1 3x+2\left(-2x-6z+3\right)-2z=2$$

Solve these equations for x and z respectively.

$$x=\frac{4}{3}-\frac{10}{3}z z=-\frac{1}{14}x+\frac{2}{7}$$

Substitute \frac{4}{3}-\frac{10}{3}z for x in the equation z=-\frac{1}{14}x+\frac{2}{7}.

$$z=-\frac{1}{14}\left(\frac{4}{3}-\frac{10}{3}z\right)+\frac{2}{7}$$

Solve z=-\frac{1}{14}\left(\frac{4}{3}-\frac{10}{3}z\right)+\frac{2}{7} for z.

$$z=\frac{1}{4}$$

Substitute \frac{1}{4} for z in the equation x=\frac{4}{3}-\frac{10}{3}z.

$$x=\frac{4}{3}-\frac{10}{3}\times \frac{1}{4}$$

Calculate x from x=\frac{4}{3}-\frac{10}{3}\times \frac{1}{4}.

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

Substitute \frac{1}{2} for x and \frac{1}{4} for z in the equation y=-2x-6z+3.

$$y=-2\times \frac{1}{2}-6\times \frac{1}{4}+3$$

Calculate y from y=-2\times \frac{1}{2}-6\times \frac{1}{4}+3.

$$y=\frac{1}{2}$$

The system is now solved.

$$x=\frac{1}{2} y=\frac{1}{2} z=\frac{1}{4}$$