A + X = 0,

Y + B - 117.72 = 0,

0.0606*X = 0.035*B,

G - X + F = -11.4031,

M - B - 117.72 = -5.9910,

-G*0.0325 + M*0.0893 - X*0.03246 +

B*0.0893 = 0.3454,

H - G = -2.6089,

P - M - 58.86 = 3.04501,

H*0.05 + G*0.0383 - M*0.0321 = 0.0652

asked by guest
on Nov 27, 2024 at 7:31 am



You asked:

Solve the system of equations:\[\begin{aligned}A + X &= 0\\Y + B - 117.72 &= 0\\0.0606 X &= 0.035 B\\G - X + F &= -11.4031\\M - B - 117.72 &= -5.9910\\-G \cdot 0.0325 + M \cdot 0.0893 - X \cdot 0.03246 + B \cdot 0.0893 &= 0.3454\\H - G &= -2.6089\\P - M - 58.86 &= 3.04501\\H \cdot 0.05 + G \cdot 0.0383 - M \cdot 0.0321 &= 0.0652\end{aligned}\]

MathBot Answer:

The system of equations has one solution.\[A = \frac{63663760279}{1980366120}\] \[B = - \frac{918577112597}{16503051000}\] \[F = - \frac{32749724874013}{495091530000}\] \[G = \frac{31078351607}{1375254250}\] \[H = \frac{1099618031767}{55010170000}\] \[M = \frac{462646136291}{8251525500}\] \[P = \frac{194691380976751}{1650305100000}\] \[X = - \frac{63663760279}{1980366120}\] \[Y = \frac{2861316276317}{16503051000}\]

Solve by Gauss-Jordan Elimination:

Begin by writing the augmented matrix of the system of equations. $$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & - \frac{7}{200} & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & 0 &\bigm |& 0\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{111729}{1000}\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0 &\bigm |& \frac{1727}{5000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Then use a series of elementary row operations to convert the matrix into reduced-row echelon form. The three elementary row operations are:

  1. Swap the positions of any two rows.

  2. Multiply any row by a nonzero scalar.

  3. Multiply a row by a nonzero scalar and add it to any other row.


First, switch the rows in the matrix such that the row with the leftmost non-zero entry with the greatest magnitude is at the top.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & - \frac{7}{200} & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & 0 &\bigm |& 0\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{111729}{1000}\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0 &\bigm |& \frac{1727}{5000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

The leading term of row \(1\) is already \(1\) so this row does not need to be multiplied by a scalar.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & - \frac{7}{200} & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & 0 &\bigm |& 0\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{111729}{1000}\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0 &\bigm |& \frac{1727}{5000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Switch the rows in the matrix such that the row with the next leftmost non-zero entry with the next greatest magnitude is the next row from the top.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & - \frac{7}{200} & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & 0 &\bigm |& 0\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{111729}{1000}\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0 &\bigm |& \frac{1727}{5000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

The leading term of row \(2\) is already \(1\) so this row does not need to be multiplied by a scalar.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & - \frac{7}{200} & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & 0 &\bigm |& 0\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{111729}{1000}\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0 &\bigm |& \frac{1727}{5000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Multiply row \(2\) by scalar \(\frac{7}{200}\) and add it to row \(3\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{111729}{1000}\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0 &\bigm |& \frac{1727}{5000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Multiply row \(2\) by scalar \(1\) and add it to row \(5\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0 &\bigm |& \frac{1727}{5000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Multiply row \(2\) by scalar \(- \frac{893}{10000}\) and add it to row \(6\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2541749}{250000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Switch the rows in the matrix such that the row with the next leftmost non-zero entry with the next greatest magnitude is the next row from the top.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2541749}{250000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

The leading term of row \(3\) is already \(1\) so this row does not need to be multiplied by a scalar.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2541749}{250000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Switch the rows in the matrix such that the row with the next leftmost non-zero entry with the next greatest magnitude is the next row from the top.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &\bigm |& - \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2541749}{250000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Multiply row \(4\) by scalar \(-1\) to make the leading term \(1\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 &\bigm |& - \frac{114031}{10000}\\0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 &\bigm |& \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2541749}{250000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Multiply row \(4\) by scalar \(-1\) and add it to row \(3\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 1 & 0 & 0 & -1 & 0 &\bigm |& - \frac{3503}{250}\\0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 &\bigm |& \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2541749}{250000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Multiply row \(4\) by scalar \(\frac{13}{400}\) and add it to row \(6\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 1 & 0 & 0 & -1 & 0 &\bigm |& - \frac{3503}{250}\\0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 &\bigm |& \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & - \frac{13}{400} & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{40328827}{4000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& \frac{163}{2500}\end{bmatrix}$$

Multiply row \(4\) by scalar \(- \frac{383}{10000}\) and add it to row \(9\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 1 & 0 & 0 & -1 & 0 &\bigm |& - \frac{3503}{250}\\0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 &\bigm |& \frac{26089}{10000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & - \frac{13}{400} & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{40328827}{4000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & 0 & \frac{883}{10000} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{100000000}\end{bmatrix}$$

Switch the rows in the matrix such that the row with the next leftmost non-zero entry with the next greatest magnitude is the next row from the top.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 1 & 0 & 0 & -1 & 0 &\bigm |& - \frac{3503}{250}\\0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 &\bigm |& \frac{26089}{10000}\\0 & 0 & 0 & 0 & \frac{883}{10000} & - \frac{321}{10000} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{100000000}\\0 & 0 & 0 & 0 & - \frac{13}{400} & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{40328827}{4000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(5\) by scalar \(\frac{10000}{883}\) to make the leading term \(1\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 1 & 0 & 0 & -1 & 0 &\bigm |& - \frac{3503}{250}\\0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 &\bigm |& \frac{26089}{10000}\\0 & 0 & 0 & 0 & 1 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{8830000}\\0 & 0 & 0 & 0 & - \frac{13}{400} & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{40328827}{4000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(5\) by scalar \(-1\) and add it to row \(3\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & \frac{321}{883} & 0 & -1 & 0 &\bigm |& - \frac{120253873}{8830000}\\0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 &\bigm |& \frac{26089}{10000}\\0 & 0 & 0 & 0 & 1 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{8830000}\\0 & 0 & 0 & 0 & - \frac{13}{400} & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{40328827}{4000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(5\) by scalar \(1\) and add it to row \(4\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & \frac{321}{883} & 0 & -1 & 0 &\bigm |& - \frac{120253873}{8830000}\\0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& \frac{39129}{17660}\\0 & 0 & 0 & 0 & 1 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{8830000}\\0 & 0 & 0 & 0 & - \frac{13}{400} & \frac{893}{10000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{40328827}{4000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(5\) by scalar \(\frac{13}{400}\) and add it to row \(6\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & \frac{321}{883} & 0 & -1 & 0 &\bigm |& - \frac{120253873}{8830000}\\0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& \frac{39129}{17660}\\0 & 0 & 0 & 0 & 1 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{8830000}\\0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{8913872843}{883000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Switch the rows in the matrix such that the row with the next leftmost non-zero entry with the next greatest magnitude is the next row from the top.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & \frac{321}{883} & 0 & -1 & 0 &\bigm |& - \frac{120253873}{8830000}\\0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& \frac{39129}{17660}\\0 & 0 & 0 & 0 & 1 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{8830000}\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &\bigm |& \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{8913872843}{883000000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(6\) by scalar \(-1\) to make the leading term \(1\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & \frac{321}{883} & 0 & -1 & 0 &\bigm |& - \frac{120253873}{8830000}\\0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& \frac{39129}{17660}\\0 & 0 & 0 & 0 & 1 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{8913872843}{883000000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(6\) by scalar \(- \frac{321}{883}\) and add it to row \(3\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} & -1 & 0 &\bigm |& \frac{784612091}{88300000}\\0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& \frac{39129}{17660}\\0 & 0 & 0 & 0 & 1 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{8913872843}{883000000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(6\) by scalar \(\frac{321}{883}\) and add it to row \(4\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} & -1 & 0 &\bigm |& \frac{784612091}{88300000}\\0 & 0 & 0 & 1 & 0 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{1791505821}{88300000}\\0 & 0 & 0 & 0 & 1 & - \frac{321}{883} & 0 & 0 & 0 &\bigm |& - \frac{3472087}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{8913872843}{883000000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(6\) by scalar \(\frac{321}{883}\) and add it to row \(5\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} & -1 & 0 &\bigm |& \frac{784612091}{88300000}\\0 & 0 & 0 & 1 & 0 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{1791505821}{88300000}\\0 & 0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{2021871691}{88300000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & 0 & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{8913872843}{883000000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(6\) by scalar \(- \frac{342097}{4415000}\) and add it to row \(8\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} & -1 & 0 &\bigm |& \frac{784612091}{88300000}\\0 & 0 & 0 & 1 & 0 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{1791505821}{88300000}\\0 & 0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{2021871691}{88300000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2339184600903}{441500000000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\end{bmatrix}$$

Multiply row \(6\) by scalar \(-1\) and add it to row \(9\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} & -1 & 0 &\bigm |& \frac{784612091}{88300000}\\0 & 0 & 0 & 1 & 0 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{1791505821}{88300000}\\0 & 0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{2021871691}{88300000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2339184600903}{441500000000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\end{bmatrix}$$

Switch the rows in the matrix such that the row with the next leftmost non-zero entry with the next greatest magnitude is the next row from the top.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} & -1 & 0 &\bigm |& \frac{784612091}{88300000}\\0 & 0 & 0 & 1 & 0 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{1791505821}{88300000}\\0 & 0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{2021871691}{88300000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2339184600903}{441500000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\end{bmatrix}$$

The leading term of row \(7\) is already \(1\) so this row does not need to be multiplied by a scalar.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} & -1 & 0 &\bigm |& \frac{784612091}{88300000}\\0 & 0 & 0 & 1 & 0 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{1791505821}{88300000}\\0 & 0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{2021871691}{88300000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2339184600903}{441500000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\end{bmatrix}$$

Multiply row \(7\) by scalar \(- \frac{321}{883}\) and add it to row \(3\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & - \frac{321}{883} &\bigm |& - \frac{856785163}{8830000}\\0 & 0 & 0 & 1 & 0 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{1791505821}{88300000}\\0 & 0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{2021871691}{88300000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2339184600903}{441500000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\end{bmatrix}$$

Multiply row \(7\) by scalar \(\frac{321}{883}\) and add it to row \(4\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & - \frac{321}{883} &\bigm |& - \frac{856785163}{8830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & - \frac{321}{883} & 0 & 0 &\bigm |& - \frac{2021871691}{88300000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2339184600903}{441500000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\end{bmatrix}$$

Multiply row \(7\) by scalar \(\frac{321}{883}\) and add it to row \(5\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & - \frac{321}{883} &\bigm |& - \frac{856785163}{8830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 &\bigm |& - \frac{6190501}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2339184600903}{441500000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\end{bmatrix}$$

Multiply row \(7\) by scalar \(1\) and add it to row \(6\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & - \frac{321}{883} &\bigm |& - \frac{856785163}{8830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{342097}{4415000} & - \frac{1623}{50000} & - \frac{893}{10000} &\bigm |& - \frac{2339184600903}{441500000000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\end{bmatrix}$$

Multiply row \(7\) by scalar \(- \frac{342097}{4415000}\) and add it to row \(8\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & - \frac{321}{883} &\bigm |& - \frac{856785163}{8830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1623}{50000} & - \frac{1472713}{8830000} &\bigm |& - \frac{7691448673}{275937500}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\end{bmatrix}$$

Switch the rows in the matrix such that the row with the next leftmost non-zero entry with the next greatest magnitude is the next row from the top.

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & - \frac{321}{883} &\bigm |& - \frac{856785163}{8830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & \frac{7}{200} &\bigm |& \frac{20601}{5000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1623}{50000} & - \frac{1472713}{8830000} &\bigm |& - \frac{7691448673}{275937500}\end{bmatrix}$$

Multiply row \(8\) by scalar \(\frac{5000}{303}\) to make the leading term \(1\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & - \frac{321}{883} &\bigm |& - \frac{856785163}{8830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1623}{50000} & - \frac{1472713}{8830000} &\bigm |& - \frac{7691448673}{275937500}\end{bmatrix}$$

Multiply row \(8\) by scalar \(-1\) and add it to row \(1\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{175}{303} &\bigm |& - \frac{6867}{101}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & - \frac{321}{883} &\bigm |& - \frac{856785163}{8830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1623}{50000} & - \frac{1472713}{8830000} &\bigm |& - \frac{7691448673}{275937500}\end{bmatrix}$$

Multiply row \(8\) by scalar \(1\) and add it to row \(3\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{175}{303} &\bigm |& - \frac{6867}{101}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{57262}{267549} &\bigm |& - \frac{25899691463}{891830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1623}{50000} & - \frac{1472713}{8830000} &\bigm |& - \frac{7691448673}{275937500}\end{bmatrix}$$

Multiply row \(8\) by scalar \(\frac{1623}{50000}\) and add it to row \(9\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{175}{303} &\bigm |& - \frac{6867}{101}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{57262}{267549} &\bigm |& - \frac{25899691463}{891830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{16503051}{111478750} &\bigm |& - \frac{2861316276317}{111478750000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(- \frac{111478750}{16503051}\) to make the leading term \(1\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{175}{303} &\bigm |& - \frac{6867}{101}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{57262}{267549} &\bigm |& - \frac{25899691463}{891830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(\frac{175}{303}\) and add it to row \(1\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{63663760279}{1980366120}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2943}{25}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{57262}{267549} &\bigm |& - \frac{25899691463}{891830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(-1\) and add it to row \(2\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{63663760279}{1980366120}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{918577112597}{16503051000}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{57262}{267549} &\bigm |& - \frac{25899691463}{891830000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(- \frac{57262}{267549}\) and add it to row \(3\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{63663760279}{1980366120}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{918577112597}{16503051000}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{32749724874013}{495091530000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{75609579}{883000}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(- \frac{321}{883}\) and add it to row \(4\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{63663760279}{1980366120}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{918577112597}{16503051000}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{32749724874013}{495091530000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{31078351607}{1375254250}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{321}{883} &\bigm |& \frac{733059203}{8830000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(- \frac{321}{883}\) and add it to row \(5\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{63663760279}{1980366120}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{918577112597}{16503051000}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{32749724874013}{495091530000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{31078351607}{1375254250}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 &\bigm |& \frac{1099618031767}{55010170000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &\bigm |& \frac{229449}{1000}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(-1\) and add it to row \(6\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{63663760279}{1980366120}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{918577112597}{16503051000}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{32749724874013}{495091530000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{31078351607}{1375254250}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 &\bigm |& \frac{1099618031767}{55010170000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{462646136291}{8251525500}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &\bigm |& \frac{29135401}{100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(-1\) and add it to row \(7\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{63663760279}{1980366120}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{918577112597}{16503051000}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{32749724874013}{495091530000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{31078351607}{1375254250}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 &\bigm |& \frac{1099618031767}{55010170000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{462646136291}{8251525500}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 &\bigm |& \frac{194691380976751}{1650305100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{175}{303} &\bigm |& \frac{6867}{101}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Multiply row \(9\) by scalar \(- \frac{175}{303}\) and add it to row \(8\).

$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{63663760279}{1980366120}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{918577112597}{16503051000}\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &\bigm |& - \frac{32749724874013}{495091530000}\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 &\bigm |& \frac{31078351607}{1375254250}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 &\bigm |& \frac{1099618031767}{55010170000}\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &\bigm |& \frac{462646136291}{8251525500}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 &\bigm |& \frac{194691380976751}{1650305100000}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &\bigm |& - \frac{63663760279}{1980366120}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &\bigm |& \frac{2861316276317}{16503051000}\end{bmatrix}$$

Once the matrix is in reduced-row echelon form, convert the matrix back into linear equations to find the solution. $$\begin{aligned}1 \cdot A+ 0 \cdot B+ 0 \cdot F+ 0 \cdot G+ 0 \cdot H+ 0 \cdot M+ 0 \cdot P+ 0 \cdot X+ 0 \cdot Y = \frac{63663760279}{1980366120} \\ A = \frac{63663760279}{1980366120}\end{aligned}$$$$\begin{aligned}0 \cdot A+ 1 \cdot B+ 0 \cdot F+ 0 \cdot G+ 0 \cdot H+ 0 \cdot M+ 0 \cdot P+ 0 \cdot X+ 0 \cdot Y = - \frac{918577112597}{16503051000} \\ B = - \frac{918577112597}{16503051000}\end{aligned}$$$$\begin{aligned}0 \cdot A+ 0 \cdot B+ 1 \cdot F+ 0 \cdot G+ 0 \cdot H+ 0 \cdot M+ 0 \cdot P+ 0 \cdot X+ 0 \cdot Y = - \frac{32749724874013}{495091530000} \\ F = - \frac{32749724874013}{495091530000}\end{aligned}$$$$\begin{aligned}0 \cdot A+ 0 \cdot B+ 0 \cdot F+ 1 \cdot G+ 0 \cdot H+ 0 \cdot M+ 0 \cdot P+ 0 \cdot X+ 0 \cdot Y = \frac{31078351607}{1375254250} \\ G = \frac{31078351607}{1375254250}\end{aligned}$$$$\begin{aligned}0 \cdot A+ 0 \cdot B+ 0 \cdot F+ 0 \cdot G+ 1 \cdot H+ 0 \cdot M+ 0 \cdot P+ 0 \cdot X+ 0 \cdot Y = \frac{1099618031767}{55010170000} \\ H = \frac{1099618031767}{55010170000}\end{aligned}$$$$\begin{aligned}0 \cdot A+ 0 \cdot B+ 0 \cdot F+ 0 \cdot G+ 0 \cdot H+ 1 \cdot M+ 0 \cdot P+ 0 \cdot X+ 0 \cdot Y = \frac{462646136291}{8251525500} \\ M = \frac{462646136291}{8251525500}\end{aligned}$$$$\begin{aligned}0 \cdot A+ 0 \cdot B+ 0 \cdot F+ 0 \cdot G+ 0 \cdot H+ 0 \cdot M+ 1 \cdot P+ 0 \cdot X+ 0 \cdot Y = \frac{194691380976751}{1650305100000} \\ P = \frac{194691380976751}{1650305100000}\end{aligned}$$$$\begin{aligned}0 \cdot A+ 0 \cdot B+ 0 \cdot F+ 0 \cdot G+ 0 \cdot H+ 0 \cdot M+ 0 \cdot P+ 1 \cdot X+ 0 \cdot Y = - \frac{63663760279}{1980366120} \\ X = - \frac{63663760279}{1980366120}\end{aligned}$$$$\begin{aligned}0 \cdot A+ 0 \cdot B+ 0 \cdot F+ 0 \cdot G+ 0 \cdot H+ 0 \cdot M+ 0 \cdot P+ 0 \cdot X+ 1 \cdot Y = \frac{2861316276317}{16503051000} \\ Y = \frac{2861316276317}{16503051000}\end{aligned}$$

Solve by matrix inversion:

In cases where the coefficient matrix of the system of equations is invertible, we can use the inverse to solve the system. Use this method with care as matrix inversion can be numerically unstable for ill-conditioned matrices.


Express the linear equations in the form \(A \times X = B\) where \(A\) is the coefficient matrix, \(X\) is the matrix of unknowns, and \(B\) is the constant matrix.$$\left[\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & - \frac{7}{200} & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & 0\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0\\0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0\end{matrix}\right] \times \left[\begin{matrix}A\\B\\F\\G\\H\\M\\P\\X\\Y\end{matrix}\right] = \left[\begin{matrix}0\\\frac{2943}{25}\\0\\- \frac{114031}{10000}\\\frac{111729}{1000}\\\frac{1727}{5000}\\- \frac{26089}{10000}\\\frac{6190501}{100000}\\\frac{163}{2500}\end{matrix}\right]$$

The product of \(A\) and its inverse \(A^{-1}\) is the identity matrix. Any matrix multiplied by the identity matrix remains unchanged, so this yields the matrix of unknowns on the left hand side of the equation, and the solution matrix on the right. $$\begin{aligned} A \times X &= B\\ A^{-1} \times A \times X &= A^{-1} \times B \\ I \times X &= A^{-1} \times B \\ X &= A^{-1} \times B \end{aligned}$$

Using a computer algebra system, calculate \(A^{-1}\). $$\left[\begin{matrix}1 & 0 & - \frac{920445625}{49509153} & 0 & \frac{59866975}{198036612} & - \frac{193156250}{49509153} & \frac{7109375}{99018306} & 0 & - \frac{71093750}{49509153}\\0 & 0 & \frac{59712875}{16503051} & 0 & - \frac{34551797}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 0 & \frac{41031250}{16503051}\\0 & 0 & \frac{855322750}{49509153} & 1 & - \frac{47088851}{99018306} & \frac{71577500}{49509153} & \frac{26717375}{49509153} & 0 & - \frac{534347500}{49509153}\\0 & 0 & \frac{7235875}{5501017} & 0 & \frac{3812303}{22004068} & \frac{13508750}{5501017} & - \frac{6727125}{11002034} & 0 & \frac{67271250}{5501017}\\0 & 0 & \frac{7235875}{5501017} & 0 & \frac{3812303}{22004068} & \frac{13508750}{5501017} & \frac{4274909}{11002034} & 0 & \frac{67271250}{5501017}\\0 & 0 & \frac{59712875}{16503051} & 0 & \frac{31460407}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 0 & \frac{41031250}{16503051}\\0 & 0 & \frac{59712875}{16503051} & 0 & \frac{31460407}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 1 & \frac{41031250}{16503051}\\0 & 0 & \frac{920445625}{49509153} & 0 & - \frac{59866975}{198036612} & \frac{193156250}{49509153} & - \frac{7109375}{99018306} & 0 & \frac{71093750}{49509153}\\0 & 1 & - \frac{59712875}{16503051} & 0 & \frac{34551797}{66012204} & - \frac{111478750}{16503051} & \frac{4103125}{33006102} & 0 & - \frac{41031250}{16503051}\end{matrix}\right]$$

Multiply both sides of the equation by the inverse. $$\left[\begin{matrix}1 & 0 & - \frac{920445625}{49509153} & 0 & \frac{59866975}{198036612} & - \frac{193156250}{49509153} & \frac{7109375}{99018306} & 0 & - \frac{71093750}{49509153}\\0 & 0 & \frac{59712875}{16503051} & 0 & - \frac{34551797}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 0 & \frac{41031250}{16503051}\\0 & 0 & \frac{855322750}{49509153} & 1 & - \frac{47088851}{99018306} & \frac{71577500}{49509153} & \frac{26717375}{49509153} & 0 & - \frac{534347500}{49509153}\\0 & 0 & \frac{7235875}{5501017} & 0 & \frac{3812303}{22004068} & \frac{13508750}{5501017} & - \frac{6727125}{11002034} & 0 & \frac{67271250}{5501017}\\0 & 0 & \frac{7235875}{5501017} & 0 & \frac{3812303}{22004068} & \frac{13508750}{5501017} & \frac{4274909}{11002034} & 0 & \frac{67271250}{5501017}\\0 & 0 & \frac{59712875}{16503051} & 0 & \frac{31460407}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 0 & \frac{41031250}{16503051}\\0 & 0 & \frac{59712875}{16503051} & 0 & \frac{31460407}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 1 & \frac{41031250}{16503051}\\0 & 0 & \frac{920445625}{49509153} & 0 & - \frac{59866975}{198036612} & \frac{193156250}{49509153} & - \frac{7109375}{99018306} & 0 & \frac{71093750}{49509153}\\0 & 1 & - \frac{59712875}{16503051} & 0 & \frac{34551797}{66012204} & - \frac{111478750}{16503051} & \frac{4103125}{33006102} & 0 & - \frac{41031250}{16503051}\end{matrix}\right] \times \left[\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & - \frac{7}{200} & 0 & 0 & 0 & 0 & 0 & \frac{303}{5000} & 0\\0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0\\0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\0 & \frac{893}{10000} & 0 & - \frac{13}{400} & 0 & \frac{893}{10000} & 0 & - \frac{1623}{50000} & 0\\0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0\\0 & 0 & 0 & \frac{383}{10000} & \frac{1}{20} & - \frac{321}{10000} & 0 & 0 & 0\end{matrix}\right] \times \left[\begin{matrix}A\\B\\F\\G\\H\\M\\P\\X\\Y\end{matrix}\right] = \left[\begin{matrix}1 & 0 & - \frac{920445625}{49509153} & 0 & \frac{59866975}{198036612} & - \frac{193156250}{49509153} & \frac{7109375}{99018306} & 0 & - \frac{71093750}{49509153}\\0 & 0 & \frac{59712875}{16503051} & 0 & - \frac{34551797}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 0 & \frac{41031250}{16503051}\\0 & 0 & \frac{855322750}{49509153} & 1 & - \frac{47088851}{99018306} & \frac{71577500}{49509153} & \frac{26717375}{49509153} & 0 & - \frac{534347500}{49509153}\\0 & 0 & \frac{7235875}{5501017} & 0 & \frac{3812303}{22004068} & \frac{13508750}{5501017} & - \frac{6727125}{11002034} & 0 & \frac{67271250}{5501017}\\0 & 0 & \frac{7235875}{5501017} & 0 & \frac{3812303}{22004068} & \frac{13508750}{5501017} & \frac{4274909}{11002034} & 0 & \frac{67271250}{5501017}\\0 & 0 & \frac{59712875}{16503051} & 0 & \frac{31460407}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 0 & \frac{41031250}{16503051}\\0 & 0 & \frac{59712875}{16503051} & 0 & \frac{31460407}{66012204} & \frac{111478750}{16503051} & - \frac{4103125}{33006102} & 1 & \frac{41031250}{16503051}\\0 & 0 & \frac{920445625}{49509153} & 0 & - \frac{59866975}{198036612} & \frac{193156250}{49509153} & - \frac{7109375}{99018306} & 0 & \frac{71093750}{49509153}\\0 & 1 & - \frac{59712875}{16503051} & 0 & \frac{34551797}{66012204} & - \frac{111478750}{16503051} & \frac{4103125}{33006102} & 0 & - \frac{41031250}{16503051}\end{matrix}\right] \times \left[\begin{matrix}0\\\frac{2943}{25}\\0\\- \frac{114031}{10000}\\\frac{111729}{1000}\\\frac{1727}{5000}\\- \frac{26089}{10000}\\\frac{6190501}{100000}\\\frac{163}{2500}\end{matrix}\right]$$ $$\left[\begin{matrix}A\\B\\F\\G\\H\\M\\P\\X\\Y\end{matrix}\right] = \left[\begin{matrix}\frac{63663760279}{1980366120}\\- \frac{918577112597}{16503051000}\\- \frac{32749724874013}{495091530000}\\\frac{31078351607}{1375254250}\\\frac{1099618031767}{55010170000}\\\frac{462646136291}{8251525500}\\\frac{194691380976751}{1650305100000}\\- \frac{63663760279}{1980366120}\\\frac{2861316276317}{16503051000}\end{matrix}\right]$$

\[A = \frac{63663760279}{1980366120}\]\[B = - \frac{918577112597}{16503051000}\]\[F = - \frac{32749724874013}{495091530000}\]\[G = \frac{31078351607}{1375254250}\]\[H = \frac{1099618031767}{55010170000}\]\[M = \frac{462646136291}{8251525500}\]\[P = \frac{194691380976751}{1650305100000}\]\[X = - \frac{63663760279}{1980366120}\]\[Y = \frac{2861316276317}{16503051000}\]

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