Blended with equations

For discussing Olympiad Level Algebra (and Inequality) problems
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ahsaf
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Blended with equations

Unread post by ahsaf » Thu Dec 22, 2016 12:01 am

$ab + bc$ $= 130$
$bc + ca$ $= 168$
$ca + ab$ $= 228$
Find the value of $a + b + c$ from the given set of equations
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Kazi_Zareer
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Re: Blended with equations

Unread post by Kazi_Zareer » Fri Dec 23, 2016 9:18 pm

Given that,
$ab+bc =130 ... ... ... ... ...$ (i)
$bc+ca =168 ... ... ... ... ...$ (ii)
$ca+ab =228 ... ... ... ... ...$ (iii)

Sum this three equation and you will get,

$ 2 (ab + bc + ca ) = 586 $

$\Rightarrow ab + bc + ca = 263 ... ... ... ... ...$ (iv)

From (i) and (iv) we get,
$ 130 + ca = 263 $ $[\because ab+bc =130]$

$\Rightarrow ca = 263 - 130 = 133 = 7 \times 19 $


Again, from (ii) and (iv) we get,
$ ab + 168 = 263 $ $[\because bc + ca =168]$

$\Rightarrow ab = 263 - 168 = 95 = 5 \times 19 $

Now look at this, $ ca = 7 \times 19 $ and $ ab = 5 \times 19 $ . Now the rest goes up to you. :)
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