If the system of equation `x - 2y + 5z = 3`
`2x - y + z = 1` and `11x - 7y + pz = q` has infinitely many solution, then

Options

A. `p + q = 2`
B. `p + q = 10`
C. `p-q=2`
D. `p-q=5`

Answer

To solve the problem, we need to analyze the system of equations given and determine the conditions under which it has infinitely many solutions.

The equations are:
1. \( x - 2y + 5z = 3 \) (Equation 1)
2. \( 2x - y + z = 1 \) (Equation 2)
3. \( 11x - 7y + pz = q \) (Equation 3)

For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero.

### Step 1: Write the coefficient matrix
The coefficient matrix \( A \) for the system of equations is:
\[ A = \begin{bmatrix} 1 & -2 & 5 \\ 2 & -1 & 1 \\ 11 & -7 & p \end{bmatrix} \]

### Step 2: Calculate the determinant of the matrix
The determinant \( D \) of matrix \( A \) can be calculated using the formula for the determinant of a 3x3 matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Where:
- \( a = 1, b = -2, c = 5 \)
- \( d = 2, e = -1, f = 1 \)
- \( g = 11, h = -7, i = p \)

Substituting these values into the determinant formula:
\[ D = 1((-1)p - (1)(-7)) - (-2)(2p - 11) + 5(2(-7) - (-1)(11)) \]
\[ D = 1(-p + 7) + 2(2p - 11) + 5(-14 + 11) \]
\[ D = -p + 7 + 4p - 22 - 15 \]
\[ D = 3p - 30 \]

### Step 3: Set the determinant to zero
For the system to have infinitely many solutions, we set the determinant equal to zero:
\[ 3p - 30 = 0 \]
Solving for \( p \):
\[ 3p = 30 \\ p = 10 \]

### Step 4: Find the value of \( q \)
Next, we need to find \( q \) such that the third equation also leads to infinitely many solutions. We will calculate the determinant of the augmented matrix formed by adding the constant terms from the equations.

The augmented matrix is:
\[ \begin{bmatrix} 1 & -2 & 5 & 3 \\ 2 & -1 & 1 & 1 \\ 11 & -7 & p & q \end{bmatrix} \]

### Step 5: Calculate the determinant of the augmented matrix
Let’s denote this determinant as \( D_3 \):
\[ D_3 = \begin{vmatrix} 1 & -2 & 5 & 3 \\ 2 & -1 & 1 & 1 \\ 11 & -7 & 10 & q \end{vmatrix} \]

Calculating \( D_3 \):
\[ D_3 = 1((-1)(10) - (1)(-7)) - (-2)(2(10) - 11) + 5(2(-7) - (-1)(q)) \]
\[ D_3 = 1(-10 + 7) + 2(20 - 11) + 5(-14 + q) \]
\[ D_3 = -3 + 2(9) + 5(-14 + q) \]
\[ D_3 = -3 + 18 - 70 + 5q \]
\[ D_3 = 5q - 55 \]

### Step 6: Set the determinant to zero
For infinitely many solutions, we set:
\[ 5q - 55 = 0 \]
Solving for \( q \):
\[ 5q = 55 \\ q = 11 \]

### Step 7: Find \( p - q \)
Now we can find \( p - q \):
\[ p - q = 10 - 11 = -1 \]

### Final Answer
Thus, the value of \( p - q \) is \( -1 \).