Solve {l}{x+1/2y+1/3z=1}{1/2x+1/3y+frac{1}{4}z=0}{frac{1}{3}x+frac{1}{4}y+frac{1}{5}z=0}

$$\left. \begin{array} { l } { x + \frac { 1 } { 2 } y + \frac { 1 } { 3 } z = 1 } \\ { \frac { 1 } { 2 } x + \frac { 1 } { 3 } y + \frac { 1 } { 4 } z = 0 } \\ { \frac { 1 } { 3 } x + \frac { 1 } { 4 } y + \frac { 1 } { 5 } z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=9$$

$$y=-36$$

$$z=30$$

Short Steps Using Substitution

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

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

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

$$\frac{1}{2}\left(-\frac{1}{2}y-\frac{1}{3}z+1\right)+\frac{1}{3}y+\frac{1}{4}z=0 \frac{1}{3}\left(-\frac{1}{2}y-\frac{1}{3}z+1\right)+\frac{1}{4}y+\frac{1}{5}z=0$$

Solve these equations for y and z respectively.

$$y=-6-z z=-\frac{15}{4}-\frac{15}{16}y$$

Substitute -6-z for y in the equation z=-\frac{15}{4}-\frac{15}{16}y.

$$z=-\frac{15}{4}-\frac{15}{16}\left(-6-z\right)$$

Solve z=-\frac{15}{4}-\frac{15}{16}\left(-6-z\right) for z.

$$z=30$$

Substitute 30 for z in the equation y=-6-z.

$$y=-6-30$$

Calculate y from y=-6-30.

$$y=-36$$

Substitute -36 for y and 30 for z in the equation x=-\frac{1}{2}y-\frac{1}{3}z+1.

$$x=-\frac{1}{2}\left(-36\right)-\frac{1}{3}\times 30+1$$

Calculate x from x=-\frac{1}{2}\left(-36\right)-\frac{1}{3}\times 30+1.

$$x=9$$

The system is now solved.

$$x=9 y=-36 z=30$$