Properties of equality

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Why is this relation transitive? $R = \{(x,y)\in\Bbb N^2\mid x + 4y = 10\}$

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Assume x + 4y = 10 and y + 4z = 10. Does x + 4z = 10?
Exercise. Prove for x,y in N, x + 4y = 10, the solutions are
(2,2), (6,1).

Thus if aRb and bRc, then b = 2.
Whence a = b = c = 2, so aRc.