For what value of a does the equations ax + y = 3 , 2x - 3y = 5 has no solution?

Given:

The system of equations is:

1. ax + y = 3

2. 2x - 3y = 5

Formula used:

For a system of linear equations to have no solution, the lines represented by the equations must be parallel. The condition for parallel lines is that the ratios of the coefficients of x and y must be equal, but the constant terms must be different. This condition can be written as:

For two equations of the form:

1. A₁x + B₁y = C₁

2. A₂x + B₂y = C₂

The condition for no solution is:

\(\dfrac{A_1}{A_2} = \dfrac{B_1}{B_2} \neq \dfrac{C_1}{C_2}\)

Calculations:

For the given equations:

1. ax + y = 3 (Here, A₁ = a, B₁ = 1, C₁ = 3)

2. 2x - 3y = 5 (Here, A₂ = 2, B₂ = -3, C₂ = 5)

Now, for the lines to be parallel, we apply the condition:

\(\dfrac{a}{2} = \dfrac{1}{-3}\)

⇒ a = 2 × (-1/3) = -2/3

∴ The value of a for which the system has no solution is a = -2/3.