Solve for x,y and z in the following AP 3,x,y,z,19?

To solve for x, y, and z in the arithmetic progression (AP) 3, x, y, z, 19, we need to find the common difference between consecutive terms.

The common difference (d) can be found by subtracting any two consecutive terms. Let's subtract the second term (x) from the first term (3):

d = x - 3

Similarly, we can subtract the third term (y) from the second term (x):

d = y - x

And we can subtract the fourth term (z) from the third term (y):

d = z - y

Since the common difference is the same for all three equations, we can set them equal to each other:

x - 3 = y - x = z - y

Now, let's solve these equations one by one:

1) x - 3 = y - x
2x = y + 3

2) y - x = z - y
2y = x + z

3) x - 3 = z - y
x + y = z + 3

Now, we have a system of three equations with three variables (x, y, and z). We can solve this system to find the values of x, y, and z.

From equation 2, we can express x in terms of y and z:
x = 2y - z

Substituting this expression for x in equation 1, we get:
2(2y - z) = y + 3
4y - 2z = y + 3
3y - 2z = 3

From equation 3, we can express z in terms of x and y:
z = x + y - 3

Substituting this expression for z in equation 2, we get:
2y = (2y - z) + z
2y = 2y

This equation is true for any value of y, which means y can take any value.

Therefore, there are infinitely many solutions for x, y, and z in the given AP.