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

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

Solve for x, y, z

$$x=\frac{3}{4}=0.75$$

$$y=\frac{5}{8}=0.625$$

$$z=-1$$

Short Steps Using Substitution

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

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

Substitute -2y-z+1 for x in the second and third equation.

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

Solve these equations for y and z respectively.

$$y=\frac{1}{8}-\frac{1}{2}z z=-4y+\frac{3}{2}$$

Substitute \frac{1}{8}-\frac{1}{2}z for y in the equation z=-4y+\frac{3}{2}.

$$z=-4\left(\frac{1}{8}-\frac{1}{2}z\right)+\frac{3}{2}$$

Solve z=-4\left(\frac{1}{8}-\frac{1}{2}z\right)+\frac{3}{2} for z.

$$z=-1$$

Substitute -1 for z in the equation y=\frac{1}{8}-\frac{1}{2}z.

$$y=\frac{1}{8}-\frac{1}{2}\left(-1\right)$$

Calculate y from y=\frac{1}{8}-\frac{1}{2}\left(-1\right).

$$y=\frac{5}{8}$$

Substitute \frac{5}{8} for y and -1 for z in the equation x=-2y-z+1.

$$x=-2\times \frac{5}{8}-\left(-1\right)+1$$

Calculate x from x=-2\times \frac{5}{8}-\left(-1\right)+1.

$$x=\frac{3}{4}$$

The system is now solved.

$$x=\frac{3}{4} y=\frac{5}{8} z=-1$$