Solve {l}{3x+2y+2=39}{2x+3y+2=24}{x+2y+3z=26}

$$\left\{ \begin{array} { l } { 3 x + 2 y + 2 = 39 } \\ { 2 x + 3 y + 2 = 24 } \\ { x + 2 y + 3 z = 26 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{67}{5} = 13\frac{2}{5} = 13.4$$

$$y = -\frac{8}{5} = -1\frac{3}{5} = -1.6$$

$$z = \frac{79}{15} = 5\frac{4}{15} \approx 5.266666667$$

Short Steps Using Substitution

Reorder the equations.

$$x+2y+3z=26 2x+3y+2=24 3x+2y+2=39$$

Solve x+2y+3z=26 for x.

$$x=-2y-3z+26$$

Substitute -2y-3z+26 for x in the second and third equation.

$$2\left(-2y-3z+26\right)+3y+2=24 3\left(-2y-3z+26\right)+2y+2=39$$

Solve these equations for y and z respectively.

$$y=-6z+30 z=\frac{41}{9}-\frac{4}{9}y$$

Substitute -6z+30 for y in the equation z=\frac{41}{9}-\frac{4}{9}y.

$$z=\frac{41}{9}-\frac{4}{9}\left(-6z+30\right)$$

Solve z=\frac{41}{9}-\frac{4}{9}\left(-6z+30\right) for z.

$$z=\frac{79}{15}$$

Substitute \frac{79}{15} for z in the equation y=-6z+30.

$$y=-6\times \frac{79}{15}+30$$

Calculate y from y=-6\times \frac{79}{15}+30.

$$y=-\frac{8}{5}$$

Substitute -\frac{8}{5} for y and \frac{79}{15} for z in the equation x=-2y-3z+26.

$$x=-2\left(-\frac{8}{5}\right)-3\times \frac{79}{15}+26$$

Calculate x from x=-2\left(-\frac{8}{5}\right)-3\times \frac{79}{15}+26.

$$x=\frac{67}{5}$$

The system is now solved.

$$x=\frac{67}{5} y=-\frac{8}{5} z=\frac{79}{15}$$