The ages of three brothers add up to 38 years. the oldest brother is 5 years older than the youngest brother. five years ago, the oldest was twice the age of the youngest. write and solve a system of three equations in three variables to determine the ages of the brothers.

Let y be the age of the youngest.
Let the age of the oldest be x.
x = y + 5 (1)
2(y - 5) = x - 5 (2)
Plugging the expression for x in (1) into (2), we get:
2(y - 5) = y + 5 - 5, which gives:
2y - 10 = y
y = 10
Plugging the value y = 10 into (1), we get:
x = 15.

Let the age of the 'middle' brother be m.
x + y + m = 38 (3)
Plugging the values for x and y into (3), gives:
10 + 15 + m = 38
m = 38 - (10 + 15) = 13

The brothers ages are 10, 13 and 15.