Question 1134432
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<pre>
Let start with the system from your post


    B             = 14685      (1)

    14685 + T + L = 16070      (2)

    14685 + A + L = 15580      (3)

    14685 + A + T = 15925      (4)


We have 4 independent linear equations in 4 unknown, so the solution does exist, for sure.


The system has <U>very special structure</U>, and I will show you <U>VERY SPECIAL trick</U> to easy solve it, which <U>works PERFECTLY for this structure</U>.


First, subtract the value 14685 from both sides of each equation (2), (3) and (4). You will get 


    T + L = 16070 - 14685 =  1385    (5)

    A + L = 15580 - 14685 =   895    (6)

    A + T = 15925 - 14685 =  1240    (7)


The trick starts from this point.

First, add all three the equations (5), (6) and (7). You will get


    2T + 2A + 2L = 1385 + 895 + 1240 = 3520,   or,  dividing by 2 both sides, 


    T + A + L = 1760                  (8)



Now subtract equation (5) from equation (8).  You will get

    A = 1760 - 1385 = 375.



Next, subtract equation (6) from equation (8).  You will get

    T = 1760 - 895 = 865.



Finally, subtract equation (7) from equation (8).  You will get

    L = 1710 - 1240 = 520.



The problem is just solved.  The  <U>ANSWER</U>  is


    $375 for A;  $865  for T,  and  $520  for L.
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If you want to see other similar solved problems, &nbsp;look into my lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/The-trick-to-solve-some-word-problems-with-three-and-more-unknowns.lesson>The tricks to solve some word problems with three and more unknowns using mental math</A>

in this site.



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