Determine the values of a and b so that the following system of linear equations have infinitely many solutions:    (2a-1)x +3y - 5 = 0,       3x + (b-1)y - 2 = 0

For a system of linear equations to have infinitely many solutions, the equations must be proportional to each other. This means that the coefficients of the corresponding variables and the constants must be in the same ratio.

The system of equations is:

(2a-1)x + 3y - 5 = 0 3x + (b-1)y - 2 = 0

We can rewrite these equations in the standard form as:

(2a-1)x + 3y = 5 3x + (b-1)y = 2

Now, let's equate the ratios of the coefficients of x, the coefficients of y, and the constants on the right side of the equations:

(2a - 1) / 3 = 3 / (b - 1) = 5 / 2

From (2a - 1) / 3 = 5 / 2, we can solve for a:

2a - 1 = 5/2 * 3 2a - 1 = 7.5 2a = 8.5 a = 8.5 / 2 a = 4.25

From 3 / (b - 1) = 5 / 2, we can solve for b:

3*(2) = 5*(b - 1) 6 = 5b - 5 5b = 11 b = 11 / 5 b = 2.2

So, the values of a and b that make the system of equations have infinitely many solutions are a = 4.25 and b = 2.2.