Solve {l}{-17x+7y-7z=505}{7x+4y+2z=-65}{19x+2y+8z=-369}

$$\left. \begin{array}{l}{ - 17 x + 7 y - 7 z = 505 }\\{ 7 x + 4 y + 2 z = - 65 }\\{ 19 x + 2 y + 8 z = - 369 }\end{array} \right.$$

Solve for x, y, z

$$x=-19$$

$$y=20$$

$$z=-6$$

Short Steps Using Substitution

Solve -17x+7y-7z=505 for x.

$$x=\frac{7}{17}y-\frac{7}{17}z-\frac{505}{17}$$

Substitute \frac{7}{17}y-\frac{7}{17}z-\frac{505}{17} for x in the second and third equation.

$$7\left(\frac{7}{17}y-\frac{7}{17}z-\frac{505}{17}\right)+4y+2z=-65 19\left(\frac{7}{17}y-\frac{7}{17}z-\frac{505}{17}\right)+2y+8z=-369$$

Solve these equations for y and z respectively.

$$y=\frac{270}{13}+\frac{5}{39}z z=\frac{3322}{3}-\frac{167}{3}y$$

Substitute \frac{270}{13}+\frac{5}{39}z for y in the equation z=\frac{3322}{3}-\frac{167}{3}y.

$$z=\frac{3322}{3}-\frac{167}{3}\left(\frac{270}{13}+\frac{5}{39}z\right)$$

Solve z=\frac{3322}{3}-\frac{167}{3}\left(\frac{270}{13}+\frac{5}{39}z\right) for z.

$$z=-6$$

Substitute -6 for z in the equation y=\frac{270}{13}+\frac{5}{39}z.

$$y=\frac{270}{13}+\frac{5}{39}\left(-6\right)$$

Calculate y from y=\frac{270}{13}+\frac{5}{39}\left(-6\right).

$$y=20$$

Substitute 20 for y and -6 for z in the equation x=\frac{7}{17}y-\frac{7}{17}z-\frac{505}{17}.

$$x=\frac{7}{17}\times 20-\frac{7}{17}\left(-6\right)-\frac{505}{17}$$

Calculate x from x=\frac{7}{17}\times 20-\frac{7}{17}\left(-6\right)-\frac{505}{17}.

$$x=-19$$

The system is now solved.

$$x=-19 y=20 z=-6$$