ax + 4y = 32 and 15x + 2by = 64 have infinite solutions. Then, what is the value?

Since the two equations have infinite solutions, they must be equivalent. This means that the coefficients of x and y in both equations must be proportional to each other.

From the first equation, we have:

ax + 4y = 32

Dividing by 4, we get:

(a/4)x + y = 8

Comparing this with the second equation, we have:

15x + 2by = 64

Dividing by 2, we get:

(15/2)x + by = 32

Now, we can see that the coefficients of x in both equations are proportional:

a/4 = 15/2

Cross multiplying, we get:

2a = 60

a = 30

Now, substituting the value of a back into the first equation, we get:

30x + 4y = 32

Dividing by 2, we get:

15x + 2y = 16

Comparing this with the second equation, we have:

15x + 2by = 64

Now, we can see that the coefficients of y in both equations are proportional:

2 = b

Therefore, the value of b is 2.

The given equations are:
ax + 4y = 32 ....(i)
15x + 2by = 64 ....(ii)

To find the value of a and b such that the equations have infinite solutions, we can express one variable in terms of the other.

From (i), we have:
ax = 32 - 4y
⇒ x = (32 - 4y)/a ....(iii)

Substituting (iii) in (ii), we get:
15(32 - 4y)/a + 2by = 64
⇒ 480 - 60y + 2aby = 64a
⇒ 2aby = 64a - 480 + 60y

For the equations to have infinite solutions, the coefficients of y must be zero. Therefore, we have:
a = 0
⇒ 2b(0) = 64a - 480 + 60y
⇒ 64(0) - 480 + 60y = 0
⇒ -480 + 60y = 0
⇒ y = 8

So, the value of a is 0 and the value of b is 15.