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I thought the minus sign should not be there when we move y0 to the other side.

y2 + 5/3 y0 = 8/3 x1

We have y0 = 3/5*x0 + x1 and therefore x0 = 5/3*y0 - 5/3*x1. For the equation y2 = -x0 + x1, we therefore get

y2 = -x0 + x1 = y2 = -(5/3*y0 - 5/3*x1) + x1 = -5/3*y0 + 5/3*x1 + x1 = -5/3*y0 + 8/3*x1

So, the example solution is correct, isn't it?