For what values of `k` the set of equations
`2x- 3y + 6z - 5t = 3, y -4z + t=1`,
`4x-5y+8z-9t = k` has infinite solution and no solution.

Explanation

To solve the problem of determining the values of \( k \) for which the given system of equations has infinite solutions or no solutions, we will follow these steps:

### Given Equations:
1. \( 2x - 3y + 6z - 5t = 3 \) (Equation 1)
2. \( y - 4z + t = 1 \) (Equation 2)
3. \( 4x - 5y + 8z - 9t = k \) (Equation 3)

### Step 1: Rewrite the equations
We can express the equations in a more standard form by isolating the constant terms.

From Equation 1:
\[ 2x - 3y + 6z = 3 + 5t \]

From Equation 2:
\[ 0x + y - 4z = 1 - t \]

From Equation 3:
\[ 4x - 5y + 8z = k + 9t \]

### Step 2: Form the augmented matrix
The augmented matrix corresponding to the system of equations is:
\[ \begin{bmatrix} 2 & -3 & 6 & -5 & | & 3 \\ 0 & 1 & -4 & 1 & | & 1 \\ 4 & -5 & 8 & -9 & | & k \end{bmatrix} \]

### Step 3: Perform row operations
We will perform row operations to simplify the matrix.

1. Replace Row 3 with \( R_3 - 2R_1 \):
\[ R_3 = R_3 - 2R_1 \Rightarrow 4 - 2(2) = 0, \quad -5 - 2(-3) = 1, \quad 8 - 2(6) = -4, \quad -9 - 2(-5) = 1, \quad k - 2(3) = k - 6 \]
The new matrix becomes:
\[ \begin{bmatrix} 2 & -3 & 6 & -5 & | & 3 \\ 0 & 1 & -4 & 1 & | & 1 \\ 0 & 1 & -4 & 1 & | & k - 6 \end{bmatrix} \]

2. Now, replace Row 3 with \( R_3 - R_2 \):
\[ R_3 = R_3 - R_2 \Rightarrow 0 - 0 = 0, \quad 1 - 1 = 0, \quad -4 - (-4) = 0, \quad 1 - 1 = 0, \quad (k - 6) - 1 = k - 7 \]
The new matrix becomes:
\[ \begin{bmatrix} 2 & -3 & 6 & -5 & | & 3 \\ 0 & 1 & -4 & 1 & | & 1 \\ 0 & 0 & 0 & 0 & | & k - 7 \end{bmatrix} \]

### Step 4: Determine conditions for solutions
1. **No Solution**: For the system to have no solution, the last row must represent a contradiction, which occurs when:
\[ k - 7 \neq 0 \quad \Rightarrow \quad k \neq 7 \]

2. **Infinite Solutions**: For the system to have infinite solutions, the last row must represent a true statement, which occurs when:
\[ k - 7 = 0 \quad \Rightarrow \quad k = 7 \]

### Final Answer:
- For infinite solutions, \( k = 7 \).
- For no solution, \( k \neq 7 \).