Solve {c}{-x+1/3y=0}{4x+2y+3z=1}{1/4x+1/2z=0}

$$\left\{ \begin{array} { c } { - x + \frac { 1 } { 3 } y = 0 } \\ { 4 x + 2 y + 3 z = 1 } \\ { \frac { 1 } { 4 } x + \frac { 1 } { 2 } z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{2}{17}\approx 0.117647059$$

$$y=\frac{6}{17}\approx 0.352941176$$

$$z=-\frac{1}{17}\approx -0.058823529$$

Short Steps Using Substitution

Solve -x+\frac{1}{3}y=0 for x.

$$x=\frac{1}{3}y$$

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

$$4\times \frac{1}{3}y+2y+3z=1 \frac{1}{4}\times \frac{1}{3}y+\frac{1}{2}z=0$$

Solve these equations for y and z respectively.

$$y=-\frac{9}{10}z+\frac{3}{10} z=-\frac{1}{6}y$$

Substitute -\frac{9}{10}z+\frac{3}{10} for y in the equation z=-\frac{1}{6}y.

$$z=-\frac{1}{6}\left(-\frac{9}{10}z+\frac{3}{10}\right)$$

Solve z=-\frac{1}{6}\left(-\frac{9}{10}z+\frac{3}{10}\right) for z.

$$z=-\frac{1}{17}$$

Substitute -\frac{1}{17} for z in the equation y=-\frac{9}{10}z+\frac{3}{10}.

$$y=-\frac{9}{10}\left(-\frac{1}{17}\right)+\frac{3}{10}$$

Calculate y from y=-\frac{9}{10}\left(-\frac{1}{17}\right)+\frac{3}{10}.

$$y=\frac{6}{17}$$

Substitute \frac{6}{17} for y in the equation x=\frac{1}{3}y.

$$x=\frac{1}{3}\times \frac{6}{17}$$

Calculate x from x=\frac{1}{3}\times \frac{6}{17}.

$$x=\frac{2}{17}$$

The system is now solved.

$$x=\frac{2}{17} y=\frac{6}{17} z=-\frac{1}{17}$$