91Ó°ÊÓ

An augmented matrix represents the system of equations where the coefficients of the variables form the rows of the matrix, and the constants on the other side of the equations are represented as an additional column. For the given system, the augmented matrix will look like this:\[\left(\begin{array}{cccc|c}1 & 2 & 3 & -1 & 7 \0 & 2 & -3 & 1 & 4 \1 & -4 & 1 & 0 & 3\end{array}\right)\]

We want to get this matrix in row-echelon form, which means having it in a staircase like format with leading ones starting further to the right on each subsequent row. Starting with the first row as is, the third row can be made zero by subtracting the first row from the third row, and then we exchange row 2 and row 3:\[\left(\begin{array}{cccc|c}1 & 2 & 3 & -1 & 7 \0 & -6 & -2 & 1 & -4 \0 & 2 & -3 & 1 & 4\end{array}\right)\]Then, we can multiply the second row by -1/6 to get a leading one:\[\left(\begin{array}{cccc|c}1 & 2 & 3 & -1 & 7 \0 & 1 & 1/3 & -1/6 & 2/3 \0 & 2 & -3 & 1 & 4\end{array}\right)\]Next step is to make the second element in row 3 to zero by subtracting 2*Row 2 from Row 3:\[\left(\begin{array}{cccc|c}1 & 2 & 3 & -1 & 7 \0 & 1 & 1/3 & -1/6 & 2/3 \0 & 0 & -11/3 & 5/3 & 8/3\end{array}\right)\]Now, multiplying the third row by -3/11 to get leading one:\[\left(\begin{array}{cccc|c}1 & 2 & 3 & -1 & 7 \0 & 1 & 1/3 & -1/6 & 2/3 \0 & 0 & 1 & -5/11 & -8/11\end{array}\right)\]

The system, in its current state, is a little easier to manage and the solutions can be found by back substitution, by substituting the known variables into the previous equations. Starting from the last row, we get \( z = -8/11 \). Substituting \( z \) into the second row to find \( y \), we get \( y = 2/3 + (1/6)*(-8/11) = 56/99 \). Substituting \( y \) and \( z \) into first row to find \( x \), we get \( x = (7 - 3*(56/99) + (8/11) - 1) / 2 = 135/198 \). The equation for \( w \) can be found by substitiping the values for \( x \), \( y \), \( z \) in the last equation, we get \( w = 3 +4*(135/198) - (56/99) = 281/198 \). Hence, the solution to the system is \( w = 281/198 \), \( x = 135/198 \), \( y = 56/99 \) and \( z = -8/11 \).