Solve {l}{x-y=8}{x+y+z=3}{2x-y+z=14}

$$\left\{ \begin{array} { l } { x - y = 8 } \\ { x + y + z = 3 } \\ { 2 x - y + z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=-3$$

$$z=1$$

Short Steps Using Substitution

Solve x-y=8 for x.

$$x=y+8$$

Substitute y+8 for x in the second and third equation.

$$y+8+y+z=3 2\left(y+8\right)-y+z=14$$

Solve these equations for y and z respectively.

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

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

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

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

$$z=1$$

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

$$y=-\frac{5}{2}-\frac{1}{2}$$

Calculate y from y=-\frac{5}{2}-\frac{1}{2}.

$$y=-3$$

Substitute -3 for y in the equation x=y+8.

$$x=-3+8$$

Calculate x from x=-3+8.

$$x=5$$

The system is now solved.

$$x=5 y=-3 z=1$$