Solve {l}{x+2y+5z=6-(1)}{2x-2y-5z=1-(2)}{3x+4y-2z=4-(2)}

$$\left. \begin{array} { l } { x + 2 y + 5 z = 6 - ( 1 ) } \\ { 2 x - 2 y - 5 z = 1 - ( 2 ) } \\ { 3 x + 4 y - 2 z = 4 - ( 2 ) } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$y=-\frac{1}{9}\approx -0.111111111$$

$$z=\frac{7}{9}\approx 0.777777778$$

Short Steps Using Substitution

Solve x+2y+5z=6-1 for x.

$$x=-2y-5z+5$$

Substitute -2y-5z+5 for x in the second and third equation.

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

Solve these equations for y and z respectively.

$$y=-\frac{5}{2}z+\frac{11}{6} z=-\frac{2}{17}y+\frac{13}{17}$$

Substitute -\frac{5}{2}z+\frac{11}{6} for y in the equation z=-\frac{2}{17}y+\frac{13}{17}.

$$z=-\frac{2}{17}\left(-\frac{5}{2}z+\frac{11}{6}\right)+\frac{13}{17}$$

Solve z=-\frac{2}{17}\left(-\frac{5}{2}z+\frac{11}{6}\right)+\frac{13}{17} for z.

$$z=\frac{7}{9}$$

Substitute \frac{7}{9} for z in the equation y=-\frac{5}{2}z+\frac{11}{6}.

$$y=-\frac{5}{2}\times \frac{7}{9}+\frac{11}{6}$$

Calculate y from y=-\frac{5}{2}\times \frac{7}{9}+\frac{11}{6}.

$$y=-\frac{1}{9}$$

Substitute -\frac{1}{9} for y and \frac{7}{9} for z in the equation x=-2y-5z+5.

$$x=-2\left(-\frac{1}{9}\right)-5\times \frac{7}{9}+5$$

Calculate x from x=-2\left(-\frac{1}{9}\right)-5\times \frac{7}{9}+5.

$$x=\frac{4}{3}$$

The system is now solved.

$$x=\frac{4}{3} y=-\frac{1}{9} z=\frac{7}{9}$$