If the system of equation 3x – 2y = 8, -2ax + (a – b) y = 48 has infinitely many solutions then:

Correct option is C

Given:
The system of equations:
​$$3x - 2y = 8$$​
​$$-2ax + (a - b) y = 48$$​​
Concept Used:
For a system of linear equations to have infinitely many solutions, the equations must be dependent, meaning one equation is a scalar multiple of the other.

This implies that the ratios of the corresponding coefficients and the constant terms must be equal.
Solution:
Let the two equations be:

$$\begin{cases}3x - 2y = 8 \quad \text{(1)} \\-2a x + (a - b) y = 48 \quad \text{(2)}\end{cases}$$​​
For the system to have infinitely many solutions, the ratios of the coefficients and the constant terms must be equal:

$$\frac{3}{-2a} = \frac{-2}{a - b} = \frac{8}{48}$$ 

$$\frac{3}{-2a} = \frac{-2}{a - b}$$ 

3(a - b ) = 4a 

3a  - 3b = 4a 

a + 3b = 0 

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