The equation x+2y=3,y-2x=1 and 7x-6y+a=0 are consisten for

Options

A. a=7
B. a=1
C. a=11
D. None of these

Answer

To determine the value of \( a \) for which the equations \( x + 2y = 3 \), \( y - 2x = 1 \), and \( 7x - 6y + a = 0 \) are consistent, we will follow these steps:

### Step 1: Rewrite the equations in standard form
The equations can be rewritten as:
1. \( x + 2y - 3 = 0 \) (Equation 1)
2. \( -2x + y - 1 = 0 \) (Equation 2)
3. \( 7x - 6y + a = 0 \) (Equation 3)

### Step 2: Form the coefficient matrix
The coefficients of the variables \( x \) and \( y \) along with the constants can be arranged in a matrix form:
\[ \begin{bmatrix} 1 & 2 & -3 \\ -2 & 1 & -1 \\ 7 & -6 & a \end{bmatrix} \]

### Step 3: Set up the determinant
For the system of equations to be consistent, the determinant of the coefficient matrix must be equal to zero:
\[ \text{Det} = \begin{vmatrix} 1 & 2 & -3 \\ -2 & 1 & -1 \\ 7 & -6 & a \end{vmatrix} = 0 \]

### Step 4: Calculate the determinant
Using the determinant formula for a \( 3 \times 3 \) matrix:
\[ \text{Det} = 1 \cdot \begin{vmatrix} 1 & -1 \\ -6 & a \end{vmatrix} - 2 \cdot \begin{vmatrix} -2 & -1 \\ 7 & a \end{vmatrix} - 3 \cdot \begin{vmatrix} -2 & 1 \\ 7 & -6 \end{vmatrix} \]

Calculating each of the \( 2 \times 2 \) determinants:
1. \( \begin{vmatrix} 1 & -1 \\ -6 & a \end{vmatrix} = 1 \cdot a - (-1) \cdot (-6) = a - 6 \)
2. \( \begin{vmatrix} -2 & -1 \\ 7 & a \end{vmatrix} = -2a - (-1) \cdot 7 = -2a + 7 \)
3. \( \begin{vmatrix} -2 & 1 \\ 7 & -6 \end{vmatrix} = (-2)(-6) - (1)(7) = 12 - 7 = 5 \)

Substituting these back into the determinant:
\[ \text{Det} = 1(a - 6) - 2(-2a + 7) - 3(5) \]
\[ = a - 6 + 4a - 14 - 15 \]
\[ = 5a - 35 \]

### Step 5: Set the determinant to zero
Setting the determinant equal to zero for consistency:
\[ 5a - 35 = 0 \]

### Step 6: Solve for \( a \)
\[ 5a = 35 \implies a = \frac{35}{5} = 7 \]

### Conclusion
The value of \( a \) for which the equations are consistent is:
\[ \boxed{7} \]