X+y=1 and y+z=9 calculate the ratio of this geometric progression

1. Given equations:

x + y = 1
y + z = 9

2. In the geometric progression, we substitute \( y = rx \) and \( z = r^2x \):

x + rx = 1
x(1 + r) = 1

3. Solve for \( x \):

x = \frac{1}{1 + r}

4. Substitute \( y = rx \) and \( z = r^2x \) into the second equation:

rx + r^2x = 9
x(r + r^2) = 9

5. Using \( x = \frac{1}{1 + r} \):

\frac{1}{1 + r}(r + r^2) = 9
\frac{r(1 + r)}{1 + r} = 9
r = 9

Therefore, the common ratio \( r \) of the geometric progression is

r = 9