Let the age of Logan = x, and the age of Emma = y
the sum of the ages = 69 years
so, x + y = 69
9 years ago, Logan's age was 2 times Emma's
so, (x - 9) = 2 (y - 9)
x - 9 = 2y - 18
x = 2y - 9
Substitute with x at the first equation
So,
2y - 9 + y = 69
Solve for y
3y = 69 + 9 = 78
y = 78/3 = 26
x = 69 - 26 = 43
So, Logan's age is 43 years and Emma's age is 26 years