Solve {l}{2300x+100y-200z=46}{100x+1500y+1000z=58}{-200x+1000y+3200z=-9}

$$\left. \begin{array} { l } { 2300 x + 100 y - 200 z = 46 } \\ { 100 x + 1500 y + 1000 z = 58 } \\ { - 200 x + 1000 y + 3200 z = - 9 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{1763}{107600}\approx 0.016384758$$

$$y=\frac{5269}{107600}\approx 0.048968401$$

$$z=-\frac{1839}{107600}\approx -0.017091078$$

Short Steps Using Substitution

Solve 2300x+100y-200z=46 for x.

$$x=\frac{1}{50}-\frac{1}{23}y+\frac{2}{23}z$$

Substitute \frac{1}{50}-\frac{1}{23}y+\frac{2}{23}z for x in the second and third equation.

$$100\left(\frac{1}{50}-\frac{1}{23}y+\frac{2}{23}z\right)+1500y+1000z=58 -200\left(\frac{1}{50}-\frac{1}{23}y+\frac{2}{23}z\right)+1000y+3200z=-9$$

Solve these equations for y and z respectively.

$$y=\frac{161}{4300}-\frac{29}{43}z z=-\frac{23}{14640}-\frac{58}{183}y$$

Substitute \frac{161}{4300}-\frac{29}{43}z for y in the equation z=-\frac{23}{14640}-\frac{58}{183}y.

$$z=-\frac{23}{14640}-\frac{58}{183}\left(\frac{161}{4300}-\frac{29}{43}z\right)$$

Solve z=-\frac{23}{14640}-\frac{58}{183}\left(\frac{161}{4300}-\frac{29}{43}z\right) for z.

$$z=-\frac{1839}{107600}$$

Substitute -\frac{1839}{107600} for z in the equation y=\frac{161}{4300}-\frac{29}{43}z.

$$y=\frac{161}{4300}-\frac{29}{43}\left(-\frac{1839}{107600}\right)$$

Calculate y from y=\frac{161}{4300}-\frac{29}{43}\left(-\frac{1839}{107600}\right).

$$y=\frac{5269}{107600}$$

Substitute \frac{5269}{107600} for y and -\frac{1839}{107600} for z in the equation x=\frac{1}{50}-\frac{1}{23}y+\frac{2}{23}z.

$$x=\frac{1}{50}-\frac{1}{23}\times \frac{5269}{107600}+\frac{2}{23}\left(-\frac{1839}{107600}\right)$$

Calculate x from x=\frac{1}{50}-\frac{1}{23}\times \frac{5269}{107600}+\frac{2}{23}\left(-\frac{1839}{107600}\right).

$$x=\frac{1763}{107600}$$

The system is now solved.

$$x=\frac{1763}{107600} y=\frac{5269}{107600} z=-\frac{1839}{107600}$$