Les S the set of all real values of a for which the following system of linear equations
`ax+2y+5z=1`
`2x+y+3z=1`
`3y+7z=1`
is consistent. Then the set S is:

Options

A. an empty set
B. equal to R
C. equal to R-{1}
D. equal to {1}

Answer

To determine the set \( S \) of all real values of \( a \) for which the given system of linear equations is consistent, we need to analyze the system of equations:

1. \( ax + 2y + 5z = 1 \)
2. \( 2x + y + 3z = 1 \)
3. \( 3y + 7z = 1 \)

### Step 1: Write the system in matrix form
The system can be represented in matrix form as \( A \mathbf{x} = \mathbf{b} \), where:

\[ A = \begin{pmatrix} a & 2 & 5 \\ 2 & 1 & 3 \\ 0 & 3 & 7 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \]

### Step 2: Find the determinant of matrix \( A \)
To check for consistency, we need to find the determinant of matrix \( A \) and ensure it is not equal to zero.

\[ \text{det}(A) = a \begin{vmatrix} 1 & 3 \\ 3 & 7 \end{vmatrix} - 2 \begin{vmatrix} 2 & 3 \\ 0 & 7 \end{vmatrix} + 5 \begin{vmatrix} 2 & 1 \\ 0 & 3 \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \( \begin{vmatrix} 1 & 3 \\ 3 & 7 \end{vmatrix} = (1)(7) - (3)(3) = 7 - 9 = -2 \)

2. \( \begin{vmatrix} 2 & 3 \\ 0 & 7 \end{vmatrix} = (2)(7) - (3)(0) = 14 \)

3. \( \begin{vmatrix} 2 & 1 \\ 0 & 3 \end{vmatrix} = (2)(3) - (1)(0) = 6 \)

Substituting these back into the determinant calculation:

\[ \text{det}(A) = a(-2) - 2(14) + 5(6) \]
\[ = -2a - 28 + 30 \]
\[ = -2a + 2 \]

### Step 3: Set the determinant not equal to zero
For the system to be consistent, we require:

\[ -2a + 2 \neq 0 \]

### Step 4: Solve for \( a \)
Solving the equation:

\[ -2a + 2 = 0 \implies -2a = -2 \implies a = 1 \]

Thus, the condition for consistency is:

\[ a \neq 1 \]

### Step 5: Define the set \( S \)
The set \( S \) of all real values of \( a \) for which the system is consistent is:

\[ S = \mathbb{R} \setminus \{1\} \]

### Final Answer
The set \( S \) is all real numbers except \( 1 \):

\[ S = \{ a \in \mathbb{R} : a \neq 1 \} \]