Solve {l}{x/6+y/2+z/6=1/2}{x+y+z=5}{frac{x}{3}+frac{y}{2}+frac{z}{6}=1}

$$\left. \begin{array} { l } { \frac { x } { 6 } + \frac { y } { 2 } + \frac { z } { 6 } = \frac { 1 } { 2 } } \\ { x + y + z = 5 } \\ { \frac { x } { 3 } + \frac { y } { 2 } + \frac { z } { 6 } = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-1$$

$$z=3$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$x+3y+z=3 x+y+z=5 2x+3y+z=6$$

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

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

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

$$-3y-z+3+y+z=5 2\left(-3y-z+3\right)+3y+z=6$$

Solve these equations for y and z respectively.

$$y=-1 z=-3y$$

Substitute -1 for y in the equation z=-3y.

$$z=-3\left(-1\right)$$

Calculate z from z=-3\left(-1\right).

$$z=3$$

Substitute -1 for y and 3 for z in the equation x=-3y-z+3.

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

Calculate x from x=-3\left(-1\right)-3+3.

$$x=3$$

The system is now solved.

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