Solve {c}{x+2y+3z=82}{x-3/1=y+1/2}{x-3/1=z+1/3}

$$\left. \begin{array} { c } { x + 2 y + 3 z = 82 } \\ { \frac { x - 3 } { 1 } = \frac { y + 1 } { 2 } } \\ { \frac { x - 3 } { 1 } = \frac { z + 1 } { 3 } } \end{array} \right.$$

Solve for x, y, z

$$x=9$$

$$y=11$$

$$z=17$$

Short Steps Using Substitution

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

$$x+2y+3z=82 2x-6=y+1 3x-9=z+1$$

Solve x+2y+3z=82 for x.

$$x=-2y-3z+82$$

Substitute -2y-3z+82 for x in the second and third equation.

$$2\left(-2y-3z+82\right)-6=y+1 3\left(-2y-3z+82\right)-9=z+1$$

Solve these equations for y and z respectively.

$$y=-\frac{6}{5}z+\frac{157}{5} z=\frac{118}{5}-\frac{3}{5}y$$

Substitute -\frac{6}{5}z+\frac{157}{5} for y in the equation z=\frac{118}{5}-\frac{3}{5}y.

$$z=\frac{118}{5}-\frac{3}{5}\left(-\frac{6}{5}z+\frac{157}{5}\right)$$

Solve z=\frac{118}{5}-\frac{3}{5}\left(-\frac{6}{5}z+\frac{157}{5}\right) for z.

$$z=17$$

Substitute 17 for z in the equation y=-\frac{6}{5}z+\frac{157}{5}.

$$y=-\frac{6}{5}\times 17+\frac{157}{5}$$

Calculate y from y=-\frac{6}{5}\times 17+\frac{157}{5}.

$$y=11$$

Substitute 11 for y and 17 for z in the equation x=-2y-3z+82.

$$x=-2\times 11-3\times 17+82$$

Calculate x from x=-2\times 11-3\times 17+82.

$$x=9$$

The system is now solved.

$$x=9 y=11 z=17$$