Answer:
1. y = 4n -1
2. y = 9n -2
Step-by-step explanation:
1. For some integer n, numbers that are congruent to 2 mod 4 are ... 4n+2. Then you have ...
y + 3 = 4n +2 . . . . next, subtract 3 from both sides
y = 4n -1 . . . . values of y are natural numbers for all natural numbers n
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2. Similarly, ...
3 - y = 9n +5
3 - 9n - 5 = y . . . . . add y-9n-5
-9n -2 = y . . . . . . . . collect terms
For every integer n, there is also an integer -n. For our purpose, we're only interested in those values of n that make -9n be positive. Rather than require n be negative so -9n is positive, we can require n be positive and use the expression 9n for the multiple of 9.
We want y a natural number, so we can write this as ...
y = 9n -2 . . . . for natural numbers n
