If the system of equations` 2x + 3y = 7 and 2ax + (a + b) y = 28 `has infinitely many solutions, then which one of the following is correct ?

Options

A. a = 2b
B. b = 2a
C. a = -2b
D. b = -2a

Explanation

To determine the conditions under which the system of equations has infinitely many solutions, we need to analyze the given equations:

1. **Equations Given:**
- \( 2x + 3y = 7 \) (Equation 1)
- \( 2ax + (a + b)y = 28 \) (Equation 2)

2. **Matrix Representation:**
We can represent the system of equations in matrix form as follows:
\[ \begin{pmatrix} 2 & 3 \\ 2a & a + b \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 28 \end{pmatrix} \]

3. **Condition for Infinitely Many Solutions:**
For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero. The determinant \( D \) of the matrix is given by:
\[ D = \begin{vmatrix} 2 & 3 \\ 2a & a + b \end{vmatrix} = (2)(a + b) - (3)(2a) \]

4. **Calculating the Determinant:**
Simplifying the determinant:
\[ D = 2(a + b) - 6a = 2a + 2b - 6a = -4a + 2b \]

5. **Setting the Determinant to Zero:**
For infinitely many solutions, we set the determinant equal to zero:
\[ -4a + 2b = 0 \]

6. **Solving for \( a \):**
Rearranging the equation gives:
\[ 2b = 4a \quad \Rightarrow \quad a = \frac{b}{2} \]

7. **Conclusion:**
Thus, we find that \( a = 2b \).

**Final Answer:**
The correct option is \( a = 2b \).

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