Solve {l}{2y+1=-z-2/3x}{2x+3y-z=4}{3x+2y+z=1/3}

$$\left\{ \begin{array} { l } { 2 y + 1 = - z - \frac { 2 } { 3 } x } \\ { 2 x + 3 y - z = 4 } \\ { 3 x + 2 y + z = \frac { 1 } { 3 } } \end{array} \right.$$

Solve for y, z, x

$$x=\frac{4}{7}\approx 0.571428571$$

$$y=\frac{31}{105}\approx 0.295238095$$

$$z = -\frac{69}{35} = -1\frac{34}{35} \approx -1.971428571$$

Short Steps Using Substitution

Solve 2y+1=-z-\frac{2}{3}x for z.

$$z=-2y-1-\frac{2}{3}x$$

Substitute -2y-1-\frac{2}{3}x for z in the second and third equation.

$$2x+3y-\left(-2y-1-\frac{2}{3}x\right)=4 3x+2y-2y-1-\frac{2}{3}x=\frac{1}{3}$$

Solve these equations for y and x respectively.

$$y=\frac{3}{5}-\frac{8}{15}x x=\frac{4}{7}$$

Substitute \frac{4}{7} for x in the equation y=\frac{3}{5}-\frac{8}{15}x.

$$y=\frac{3}{5}-\frac{8}{15}\times \frac{4}{7}$$

Calculate y from y=\frac{3}{5}-\frac{8}{15}\times \frac{4}{7}.

$$y=\frac{31}{105}$$

Substitute \frac{31}{105} for y and \frac{4}{7} for x in the equation z=-2y-1-\frac{2}{3}x.

$$z=-2\times \frac{31}{105}-1-\frac{2}{3}\times \frac{4}{7}$$

Calculate z from z=-2\times \frac{31}{105}-1-\frac{2}{3}\times \frac{4}{7}.

$$z=-\frac{69}{35}$$

The system is now solved.

$$y=\frac{31}{105} z=-\frac{69}{35} x=\frac{4}{7}$$