Solve {l}{x-y+3z=13}{x+y+z=11}{2x+2y-z=7}

$$\left. \begin{array} { l } { x - y + 3 z = 13 } \\ { x + y + z = 11 } \\ { 2 x + 2 y - z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=4$$

$$z=5$$

Short Steps Using Substitution

Solve x-y+3z=13 for x.

$$x=y-3z+13$$

Substitute y-3z+13 for x in the second and third equation.

$$y-3z+13+y+z=11 2\left(y-3z+13\right)+2y-z=7$$

Solve these equations for y and z respectively.

$$y=-1+z z=\frac{4}{7}y+\frac{19}{7}$$

Substitute -1+z for y in the equation z=\frac{4}{7}y+\frac{19}{7}.

$$z=\frac{4}{7}\left(-1+z\right)+\frac{19}{7}$$

Solve z=\frac{4}{7}\left(-1+z\right)+\frac{19}{7} for z.

$$z=5$$

Substitute 5 for z in the equation y=-1+z.

$$y=-1+5$$

Calculate y from y=-1+5.

$$y=4$$

Substitute 4 for y and 5 for z in the equation x=y-3z+13.

$$x=4-3\times 5+13$$

Calculate x from x=4-3\times 5+13.

$$x=2$$

The system is now solved.

$$x=2 y=4 z=5$$