Solve {l}{x-y+z=3}{x+y-3z=-10}{2x+y-z=-6}

$$\left. \begin{array} { l } { x - y + z = 3 } \\ { x + y - 3 z = - 10 } \\ { 2 x + y - z = - 6 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$z = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Short Steps Using Substitution

Solve x-y+z=3 for x.

$$x=y-z+3$$

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

$$y-z+3+y-3z=-10 2\left(y-z+3\right)+y-z=-6$$

Solve these equations for y and z respectively.

$$y=2z-\frac{13}{2} z=y+4$$

Substitute 2z-\frac{13}{2} for y in the equation z=y+4.

$$z=2z-\frac{13}{2}+4$$

Solve z=2z-\frac{13}{2}+4 for z.

$$z=\frac{5}{2}$$

Substitute \frac{5}{2} for z in the equation y=2z-\frac{13}{2}.

$$y=2\times \frac{5}{2}-\frac{13}{2}$$

Calculate y from y=2\times \frac{5}{2}-\frac{13}{2}.

$$y=-\frac{3}{2}$$

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

$$x=-\frac{3}{2}-\frac{5}{2}+3$$

Calculate x from x=-\frac{3}{2}-\frac{5}{2}+3.

$$x=-1$$

The system is now solved.

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