Two equations "combined" don't give the desired result

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The discussion addresses a potential typo in the Lorentz transformation equation, specifically noting that x' should not be in the denominator of equation (5.6.7). A participant expresses confusion over how to combine this corrected equation with another to derive a relation between time variables t and t'. Initial attempts at substitution and algebraic manipulation resulted in complications, leading to uncertainty about the derivation process. After receiving clarification, the participant realizes a minor mistake in their calculations and successfully resolves the issue. The interaction highlights the importance of careful algebraic manipulation in understanding complex physics equations.
nomadreid
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Homework Statement
From only (A) x'=g(x-vt) and (B) x^2-(ct)^2=x'^2-(ct')^2 derive
(C) t'=g(t-(vx/c^2)),
Relevant Equations
g= gamma = 1/sqrt(1-(v/c)^2) and the equations in the Statement
In https://phys.libretexts.org/Bookshe...__Relativity/5.06:_The_Lorentz_Transformation

First, the equation (5.6.7) apparently has a typo: the x' should not be in the denominator, as one can see by comparing it with the equation just above it from which it was derived. The corrected equation is Equation (A) in the Statement (standard Lorentz transformation).

Then two equations down (unnumbered), the author states the equation (B) in the Statement,
"We combine this with Equation 5.6.7 that relates x and x' to obtain the relation between t and t′:"
and then states the equation (C) in the statement.

How he means to "combine" them is what I don't successfully get. I tried substitution of x' from (A) into (B), and got a mess; I then tried solving (B) for x', and substituting this solution into (A), and got the same mess, that is,
(A) into (B)

first mess.PNG

which doesn't simplify to (C). Either: (a) my algebraic manipulation is wrong; (b) the author is including some other equation in the derivation.
Any indications where this is going wrong would be greatly appreciated. (Yes, there are other ways to derive the relation (C), but I am interested in this author's derivation.)
 
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nomadreid said:
which doesn't simplify to (C)
Yes it does (up to the sign of the square root).
 
Thanks, Orodruin. OK, I will try again, now with the assurance that I just made some minor mistake made in simplifying. That fully answers my question! Super!
 
update: found the error. It all comes out. Thanks again, Orodruin
 
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