Solve {l}{x+y+z=15}{x-y+z=5}{x-y-z=-9}

$$\left\{ \begin{array} { l } { x + y + z = 15 } \\ { x - y + z = 5 } \\ { x - y - z = - 9 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=5$$

$$z=7$$

Short Steps Using Substitution

Solve x+y+z=15 for x.

$$x=-y-z+15$$

Substitute -y-z+15 for x in the second and third equation.

$$-y-z+15-y+z=5 -y-z+15-y-z=-9$$

Solve these equations for y and z respectively.

$$y=5 z=12-y$$

Substitute 5 for y in the equation z=12-y.

$$z=12-5$$

Calculate z from z=12-5.

$$z=7$$

Substitute 5 for y and 7 for z in the equation x=-y-z+15.

$$x=-5-7+15$$

Calculate x from x=-5-7+15.

$$x=3$$

The system is now solved.

$$x=3 y=5 z=7$$