Macy, sydmey, and allie are sisters. macy is two years older than sydney and allie is one year less than half the age of sydney. if the sum of their ages is 31, how old is macy

Macy is 14.

Let s be Sydney's age.  Macy is 2 years older than Sydney, or s+2.  Allie is 1 year less than half of Sydney's age, or 1/2s-1.  Together, their combined ages are 31:
s+(s+2)+(1/2-1)=31

Combining like terms, we have
(2 1/2)s+1=31

Subtract 1 from both sides:
(2 1/2)s+1-1=31-1
(2 1/2)s = 30

Divide both by 2 1/2:
(2 1/2)s/(2 1/2) = 30÷(2 1/2)
s = 12

Sydney is 12; Macy is 12+2=14.