If the system of equations `2x-3y+5z=12, 3x_y+pz=q` and `x-7y+8z-17` is consistent, then which of the following is not true?

Options

A. `p=2, q=7`
B. `p ne 2, q=7`
C. `p ne2, q ne7`
D. `p =2, q ne 7`

Answer

To determine which statement is not true regarding the consistency of the given system of equations, we will analyze the equations using Cramer's rule. The system of equations is:

1. \( 2x - 3y + 5z = 12 \)
2. \( 3x - y + pz = q \)
3. \( x - 7y + 8z = 17 \)

### Step 1: Formulate the Coefficient Matrix and Calculate the Determinant (Δ)

The coefficient matrix \( A \) for the system can be represented as:

\[ A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & -1 & p \\ 1 & -7 & 8 \end{bmatrix} \]

We need to calculate the determinant \( \Delta \) of this matrix. The determinant is calculated as follows:

\[ \Delta = 2 \begin{vmatrix} -1 & p \\ -7 & 8 \end{vmatrix} - (-3) \begin{vmatrix} 3 & p \\ 1 & 8 \end{vmatrix} + 5 \begin{vmatrix} 3 & -1 \\ 1 & -7 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} -1 & p \\ -7 & 8 \end{vmatrix} = (-1)(8) - (-7)(p) = -8 + 7p = 7p - 8 \)

2. \( \begin{vmatrix} 3 & p \\ 1 & 8 \end{vmatrix} = (3)(8) - (1)(p) = 24 - p \)

3. \( \begin{vmatrix} 3 & -1 \\ 1 & -7 \end{vmatrix} = (3)(-7) - (-1)(1) = -21 + 1 = -20 \)

Now substituting back into the determinant formula:

\[ \Delta = 2(7p - 8) + 3(24 - p) + 5(-20) \]

Expanding this:

\[ \Delta = 14p - 16 + 72 - 3p - 100 \]

Combining like terms:

\[ \Delta = 11p - 44 \]

### Step 2: Conditions for Consistency

For the system to be consistent, the determinant \( \Delta \) must not be equal to zero for a unique solution. Therefore, we set:

\[ 11p - 44 \neq 0 \implies p \neq 4 \]

### Step 3: Calculate \( \Delta_1 \) and \( \Delta_2 \)

Next, we need to calculate \( \Delta_1 \) and \( \Delta_2 \) to check for infinite solutions.

**For \( \Delta_1 \)** (replace the first column with the RHS):

\[ \Delta_1 = \begin{vmatrix} 12 & -3 & 5 \\ q & -1 & p \\ 17 & -7 & 8 \end{vmatrix} \]

Calculating this determinant will yield a condition involving \( q \) and \( p \).

**For \( \Delta_2 \)** (replace the second column with the RHS):

\[ \Delta_2 = \begin{vmatrix} 2 & 12 & 5 \\ 3 & q & p \\ 1 & 17 & 8 \end{vmatrix} \]

Calculating this determinant will yield another condition involving \( q \) and \( p \).

### Step 4: Conclusion

The conditions for consistency can be summarized as follows:

1. \( p \neq 4 \) for a unique solution.
2. If \( \Delta = 0 \), then at least one of \( \Delta_1 \) or \( \Delta_2 \) must also be zero for infinite solutions.

### Final Answer

The statement that is **not true** is the one that contradicts the conditions derived above.