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Take $$ \frac{x-a}{b-a}(b) + \frac{b-x}{b-a}(a) = x.$$
I need an interpretation of this, using concrete example.

well i don't have difficulty doing it and is not HW. I'm ust drawing a blank right now trying to interpret this. example would be on a number like 2-9 i want 6 then applying this $(4/7)(9)+(3/7)2 =6$. I'm TRYING to see what's going on in , not making sense to me but it works

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    well i dont have difficulty doing it and is not HW. IM ust drawing a blank right now trying to interpret this. example would be on a number like 2-9 i want 6 then applying this (4/7)(9)+(3/7)2 =6 IM TRYING to see whats going on in , not making sense to me but it works,2011-02-07
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    How about putting all that in the question, instead of simply telling us what you need or want, and expect people to jump to it?2011-02-07
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    can u answer for me please ??? im just blanking out2011-02-07
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    Honestly, I have no idea what you are looking for, so no, I cannot. Your "example" didn't tell me anything. The equality holds by trivial algebra, and has nothing to do with number theory.2011-02-07
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    @kai cheung: This is some kind of weighted average with normalized weights. Note that, for any $a \leq x \leq b$, $\frac{{x - a}}{{b - a}} + \frac{{b - x}}{{b - a}} = 1$ (the weights sum up to $1$). In fact, the solution $p$ of $pb+(1-p)a=x$ is given by $p=\frac{{x - a}}{{b - a}}$ (and hence $1-p = \frac{{b - x}}{{b - a}}$).2011-02-07
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    shai thank u, i really like this.2011-02-07
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    Some intuition. You want to represent $6$ as a weighted average of the endpoints $2$ and $9$. As I commented above, the solution $p$ of $p9 + (1-p)2 = 6$ is given by $p=\frac{{6 - 2}}{{9 - 2}}=4/7$. Hence the decomposition $(4/7)9 + (3/7)2 = 6$. (Since $9-2=7$, it is natural to represent the weights as $p = t/7$ and $1-p = (7-t)/7$, where $0 < t < 7$.) For the intuition, replace $6$ by $8.5$. Suppose that $p9+(1-p)2=8.5$. Then, $p$ should be relatively close to $1$; it is not surprising that $p=(8.5-2)/(9-2)=6.5/7$. I hope that my intention is clear (next, consider $x=9$).2011-02-07
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    It's the linear interpolant passing through $(a,a)$ and $(b,b)$. I'm not sure why you think there's something deeper into this?2011-05-03

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