If the simultaneous equations ax + 4y = 32 and 15x + 2by = 64 have infinitely many solutions, then the value of a + b is _____.

To have infinitely many solutions, the two equations must be equivalent. This means that the ratios of the coefficients of x and y in both equations must be equal.

From the first equation, we have a/4 = 15/2b
Cross multiplying, we get 2ab = 60
ab = 30

Since the equations have infinitely many solutions, a and b can take on any values as long as they satisfy the above equation. One possible solution is a = 6 and b = 5.

Therefore, the value of a + b is 6 + 5 = 11.

For infinitely many solutions, the equations must be dependent.

Compare the given equations with ax + 4y = 32 and 15x + 2by = 64, we get
a1/a2 = 4/15 and b1/b2 = 4/2b = 2/b

So, a1/a2 = b1/b2 = 4/15 = 2/b
=> b = 15/2
=> a/a2 = 4/15 = b1/b2 = 2/15/2 = 4/15
=> a = 4a2/15

Therefore, a/b = (4a2/15)/(15/2) = 8a2/225

Y= ½(×+4)²+6