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

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

Solve for x, y, z

$$x=1$$

$$y=1$$

$$z=1$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for x and z respectively.

$$x=\frac{2}{7}z+\frac{5}{7} z=\frac{1}{2}+\frac{1}{2}x$$

Substitute \frac{2}{7}z+\frac{5}{7} for x in the equation z=\frac{1}{2}+\frac{1}{2}x.

$$z=\frac{1}{2}+\frac{1}{2}\left(\frac{2}{7}z+\frac{5}{7}\right)$$

Solve z=\frac{1}{2}+\frac{1}{2}\left(\frac{2}{7}z+\frac{5}{7}\right) for z.

$$z=1$$

Substitute 1 for z in the equation x=\frac{2}{7}z+\frac{5}{7}.

$$x=\frac{2}{7}\times 1+\frac{5}{7}$$

Calculate x from x=\frac{2}{7}\times 1+\frac{5}{7}.

$$x=1$$

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

$$y=-2+1+2$$

Calculate y from y=-2+1+2.

$$y=1$$

The system is now solved.

$$x=1 y=1 z=1$$