Question

2. Solve the augmented system (a) \( \left[\begin{array}{rrr:r}3 & -4 & -2 & 3 \\ 6 & -6 & 1 & 2 \\ -3 & 8 & 2 & -1\end{array}\right] \) (b) \( \left[\begin{array}{lll:l}2 & 1 & -1 & 2 \\ 6 & 2 & -2 &

Answer 1

(a) The solution to the augmented system
`\( \begin{bmatrix}3 & -4 & -2 \\ 6 & -6 & 1 \\ -3 & 8 & 2\end{bmatrix} \) is \( x = 1, y = -2, z = 3 \).`

(b) The solution to the augmented system
`\( \begin{bmatrix}2 & 1 & -1 \\ 6 & 2 & -2\end{bmatrix} \)` is inconsistent, indicating no solution exists.

Step-by-step explanation:

In the first augmented system, we can perform row operations to transform the matrix into its reduced row-echelon form. After applying Gaussian elimination, the matrix becomes
`\( \begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & -2 \\ 0 & 0 & 0\end{bmatrix} \),` revealing that
`\( x = 1 \), \( y = -2 \), and \( z = 3 \).`

Now, turning to the second augmented system, after applying row operations, the matrix transforms into
`\( \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 1\end{bmatrix} \),` which implies ( x = 0 ), ( y = 1 ), and ( z ) is a free variable. However, the inconsistency arises from the discrepancy in the last column. The system is inconsistent as the right side of the system does not match the coefficients. This signifies that no unique solution exists for this system.

In summary, the first system has a unique solution ( x = 1, y = -2, z = 3 ), while the second system is inconsistent, indicating no solution. These results are obtained through the application of systematic row operations and the interpretation of the augmented matrices.

Answer 2

(a) The augmented system
`\( \left[\begin{array}{rrr:r}3 & -4 & -2 & 3 \\ 6 & -6 & 1 & 2 \\ -3 & 8 & 2 & -1\end{array}\right] \)` is consistent and has a unique solution. The solution is
`\( x = 1, y = 2, z = -1 \).`

(b) The augmented system
`\( \left[\begin{array}{lll:l}2 & 1 & -1 & 2 \\ 6 & 2 & -2 & \end{array}\right] \)` is inconsistent and does not have a solution.

Step-by-step explanation:

For part (a), let's denote the system as follows:

[3x - 4y - 2z = 3]

[6x - 6y + z = 2]

[-3x + 8y + 2z = -1]

We can solve this system using the Gaussian elimination method. Performing row operations, we obtain the following system:

[x + 0y + 0z = 1]

[0x + y + 0z = 2]

[0x + 0y + z = -1]

This simplified system directly gives us the values of the variables: (x = 1, y = 2, z = -1). Therefore, the solution to the augmented system is consistent and unique.

For part (b), the augmented system can be denoted as:

2x + y - z = 2

6x + 2y - 2z = k

Performing Gaussian elimination, we reach a point where the system is inconsistent as it leads to a contradiction. This means there is no set of values for (x, y, z) that satisfies both equations simultaneously. Therefore, the system is inconsistent and does not have a solution.