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

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

Solve for x, y, z

$$x=-2$$

$$y=3$$

$$z=1$$

Short Steps Using Substitution

Solve \frac{1}{2}x+\frac{1}{3}y=0 for x.

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

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

$$4\left(-\frac{2}{3}\right)y+2y+3z=1 \frac{1}{4}\left(-\frac{2}{3}\right)y+\frac{1}{2}z=0$$

Solve these equations for y and z respectively.

$$y=\frac{9}{2}z-\frac{3}{2} z=\frac{1}{3}y$$

Substitute \frac{9}{2}z-\frac{3}{2} for y in the equation z=\frac{1}{3}y.

$$z=\frac{1}{3}\left(\frac{9}{2}z-\frac{3}{2}\right)$$

Solve z=\frac{1}{3}\left(\frac{9}{2}z-\frac{3}{2}\right) for z.

$$z=1$$

Substitute 1 for z in the equation y=\frac{9}{2}z-\frac{3}{2}.

$$y=\frac{9}{2}\times 1-\frac{3}{2}$$

Calculate y from y=\frac{9}{2}\times 1-\frac{3}{2}.

$$y=3$$

Substitute 3 for y in the equation x=-\frac{2}{3}y.

$$x=-\frac{2}{3}\times 3$$

Calculate x from x=-\frac{2}{3}\times 3.

$$x=-2$$

The system is now solved.

$$x=-2 y=3 z=1$$