Solve {l}{4x-5y+2z=-4}{3x+5z=-1}{6x+7y-2z=11}

$$\left. \begin{array} { l } { 4 x - 5 y + 2 z = - 4 } \\ { 3 x + 5 z = - 1 } \\ { 6 x + 7 y - 2 z = 11 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{1}{2}=0.5$$

$$y=1$$

$$z=-\frac{1}{2}=-0.5$$

Short Steps Using Substitution

Solve 4x-5y+2z=-4 for x.

$$x=\frac{5}{4}y-\frac{1}{2}z-1$$

Substitute \frac{5}{4}y-\frac{1}{2}z-1 for x in the second and third equation.

$$3\left(\frac{5}{4}y-\frac{1}{2}z-1\right)+5z=-1 6\left(\frac{5}{4}y-\frac{1}{2}z-1\right)+7y-2z=11$$

Solve these equations for y and z respectively.

$$y=\frac{8}{15}-\frac{14}{15}z z=-\frac{17}{5}+\frac{29}{10}y$$

Substitute \frac{8}{15}-\frac{14}{15}z for y in the equation z=-\frac{17}{5}+\frac{29}{10}y.

$$z=-\frac{17}{5}+\frac{29}{10}\left(\frac{8}{15}-\frac{14}{15}z\right)$$

Solve z=-\frac{17}{5}+\frac{29}{10}\left(\frac{8}{15}-\frac{14}{15}z\right) for z.

$$z=-\frac{1}{2}$$

Substitute -\frac{1}{2} for z in the equation y=\frac{8}{15}-\frac{14}{15}z.

$$y=\frac{8}{15}-\frac{14}{15}\left(-\frac{1}{2}\right)$$

Calculate y from y=\frac{8}{15}-\frac{14}{15}\left(-\frac{1}{2}\right).

$$y=1$$

Substitute 1 for y and -\frac{1}{2} for z in the equation x=\frac{5}{4}y-\frac{1}{2}z-1.

$$x=\frac{5}{4}\times 1-\frac{1}{2}\left(-\frac{1}{2}\right)-1$$

Calculate x from x=\frac{5}{4}\times 1-\frac{1}{2}\left(-\frac{1}{2}\right)-1.

$$x=\frac{1}{2}$$

The system is now solved.

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