Solve {l}{x-y+z=3}{6x+9y+4z=9}{12x+6y-z=4}

$$\left\{ \begin{array} { l } { x - y + z = 3 } \\ { 6 x + 9 y + 4 z = 9 } \\ { 12 x + 6 y - z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{2}{3}\approx 0.666666667$$

$$y=-\frac{1}{3}\approx -0.333333333$$

$$z=2$$

Short Steps Using Substitution

Solve x-y+z=3 for x.

$$x=y-z+3$$

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

$$6\left(y-z+3\right)+9y+4z=9 12\left(y-z+3\right)+6y-z=4$$

Solve these equations for y and z respectively.

$$y=\frac{2}{15}z-\frac{3}{5} z=\frac{32}{13}+\frac{18}{13}y$$

Substitute \frac{2}{15}z-\frac{3}{5} for y in the equation z=\frac{32}{13}+\frac{18}{13}y.

$$z=\frac{32}{13}+\frac{18}{13}\left(\frac{2}{15}z-\frac{3}{5}\right)$$

Solve z=\frac{32}{13}+\frac{18}{13}\left(\frac{2}{15}z-\frac{3}{5}\right) for z.

$$z=2$$

Substitute 2 for z in the equation y=\frac{2}{15}z-\frac{3}{5}.

$$y=\frac{2}{15}\times 2-\frac{3}{5}$$

Calculate y from y=\frac{2}{15}\times 2-\frac{3}{5}.

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

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

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

Calculate x from x=-\frac{1}{3}-2+3.

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

The system is now solved.

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