I'm new to Math Induction, I don't know how it goes from $𝑘 + 1 = 3𝑥 + 5𝑦 + 1$ into $5𝑦 = 5𝑦 − 5 + 5.$  I'm trying to prove that:   Any whole number of at least 8 cents can be obtained using coins of 3 and 5 cents  Assume: There is a number $𝑘$ for which there are $𝑥$ and $𝑦$ such that $𝑘 = 3𝑥 + 5𝑦$  Here's my working:  $𝑘 = 3𝑥 + 5𝑦$  Therefore $𝑘 + 1 = 3𝑥 + 5𝑦 + 1$  $5𝑦 = 5𝑦 − 5 + 5$  $5𝑦 = 5 𝑦 − 1 + 5$  $𝑘 + 1 = 3𝑥 + 5𝑦 + 1$  $𝑘 + 1 = 3𝑥 + 5(𝑦 − 1) + 5 + 1$  $𝑘 + 1 = 3𝑥 + 5(𝑦 − 1) + 6$  $𝑘 + 1 = 3(𝑥 + 2) + 5(𝑦 − 1)$  $𝒌 + 𝟏 = 𝟑𝒙' + 𝟓𝒚'$
I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.  $$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$  I want to check whether that would be: 0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.  Meaning that it would include 0, 3, 5, 8. Then from 9 and on, every Natural Number. But how would I explain it as a set? or prove that these are the numbers in the set?
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