Solve {l}{2x+7y-6z=-74}{1x+6y+4z=9}{8x+2y+3z=24}

$$\left. \begin{array} { l } { 2 x + 7 y - 6 z = - 74 } \\ { 1 x + 6 y + 4 z = 9 } \\ { 8 x + 2 y + 3 z = 24 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-4$$

$$z=8$$

Short Steps Using Substitution

Solve 2x+7y-6z=-74 for x.

$$x=-\frac{7}{2}y+3z-37$$

Substitute -\frac{7}{2}y+3z-37 for x in the second and third equation.

$$1\left(-\frac{7}{2}y+3z-37\right)+6y+4z=9 8\left(-\frac{7}{2}y+3z-37\right)+2y+3z=24$$

Solve these equations for y and z respectively.

$$y=-\frac{14}{5}z+\frac{92}{5} z=\frac{320}{27}+\frac{26}{27}y$$

Substitute -\frac{14}{5}z+\frac{92}{5} for y in the equation z=\frac{320}{27}+\frac{26}{27}y.

$$z=\frac{320}{27}+\frac{26}{27}\left(-\frac{14}{5}z+\frac{92}{5}\right)$$

Solve z=\frac{320}{27}+\frac{26}{27}\left(-\frac{14}{5}z+\frac{92}{5}\right) for z.

$$z=8$$

Substitute 8 for z in the equation y=-\frac{14}{5}z+\frac{92}{5}.

$$y=-\frac{14}{5}\times 8+\frac{92}{5}$$

Calculate y from y=-\frac{14}{5}\times 8+\frac{92}{5}.

$$y=-4$$

Substitute -4 for y and 8 for z in the equation x=-\frac{7}{2}y+3z-37.

$$x=-\frac{7}{2}\left(-4\right)+3\times 8-37$$

Calculate x from x=-\frac{7}{2}\left(-4\right)+3\times 8-37.

$$x=1$$

The system is now solved.

$$x=1 y=-4 z=8$$