I read the proof, but I am lost in the solution to problem one.
Problem 1, on <http://www.cut-the-knot.com/blue/chinese.html>

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Solve

p1: x = 2 (mod 3)
p2: x = 3 (mod 5)
p3: x = 2 (mod 7)

From p1, x = 3t 2, for some integer t. Substituting this into p2 gives 3t = 1.
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I see that
x = 3t 2 = 5s 3, which gives,
3t - 5s = 1, but how does that yield
3t = 1 ?

Thank you for helping me to understand this.
(Note that I am only in high school, which may contribute to the lack of knowledge which has prevented me from understanding this; and for that I apologize.)

dan