Splitting equation into two pieces

Question

I am working through a book I just bought, and got stuck a bit.

The book uses a string of substitutions to transform an equation to one easier. So the final equation is this:

$T \cdot o - P \cdot o + T \cdot d - P \cdot d = f - a$

I substituted unknowns with real numbers and it holds. So far so good.

At this point, the book splits up the equation into 2 pieces of this form:

$T \cdot o - P \cdot d = f$ and $P \cdot o - T \cdot d = a$

and now neither of this holds when substituted with real numbers. Is this splitting step valid?

@ EDIT, real numbers:

$T = 4$, $o = 22$, $P = -1$, $d = -3.8$, $f = 138$, $a = 47$

@EDIT2,

If anyone is actually curious enough to explore the problem further, I'll be happy to exchange emails and the scan of the book page. You need to understand basic linear equations.

Please change your equation into LaTeX font by putting them in dollar signs. Also, is To a variable or are T and o variables and you multiply them to get To? Also, what are the real numbers you substitute? — 5xum, Jan 29 '14 at 11:20
With only these assumptions, no it's not valid. But there's probably more assumption you are not telling us. For example, what are the real numbers substituted? — Leo Azevedo, Jan 29 '14 at 11:21
In general we can say if $To - Pd = f$ and $Po - Td = a$ then $Td -Po + Td - Pd = f - a$ but you can't do this the other way round. — Warren Hill, Jan 29 '14 at 11:23
From the book. We are trying to solve for $d$. The book provided the solution but I don't know how they arrived at this number.ps. and why the down-vote? — nutship, Jan 29 '14 at 11:26

score 1 · Accepted Answer · answered Jan 29 '14 at 11:29

The only possible answer for this is:

No, it's not a logical step to arbitrarily separate the equations like the book did, without further assumptions. The second line actually has more information about the problem then the first one, so where did this information come from?

Splitting equation into two pieces

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