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

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

Solve for x, y, z

$$x=\frac{1}{7}\approx 0.142857143$$

$$y = \frac{8}{7} = 1\frac{1}{7} \approx 1.142857143$$

$$z = \frac{11}{7} = 1\frac{4}{7} \approx 1.571428571$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for y and z respectively.

$$y=\frac{3}{5}z+\frac{1}{5} z=\frac{1}{2}y+1$$

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

$$z=\frac{1}{2}\left(\frac{3}{5}z+\frac{1}{5}\right)+1$$

Solve z=\frac{1}{2}\left(\frac{3}{5}z+\frac{1}{5}\right)+1 for z.

$$z=\frac{11}{7}$$

Substitute \frac{11}{7} for z in the equation y=\frac{3}{5}z+\frac{1}{5}.

$$y=\frac{3}{5}\times \frac{11}{7}+\frac{1}{5}$$

Calculate y from y=\frac{3}{5}\times \frac{11}{7}+\frac{1}{5}.

$$y=\frac{8}{7}$$

Substitute \frac{8}{7} for y and \frac{11}{7} for z in the equation x=-3y+z+2.

$$x=-3\times \frac{8}{7}+\frac{11}{7}+2$$

Calculate x from x=-3\times \frac{8}{7}+\frac{11}{7}+2.

$$x=\frac{1}{7}$$

The system is now solved.

$$x=\frac{1}{7} y=\frac{8}{7} z=\frac{11}{7}$$