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Math Help - It's easy to see that...

  1. #1
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    It's easy to see that...

    You are given:

    (1) a + b + c + d = e + f + g + h

    (2) a + 2b + 3c + 4d = e + 2f + 3g + 4h

    Now give me back:

    (3) 4a + 3b + 2c + d = 4e + 3f + 2g + h

    (show all work in your proof, please).
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  2. #2
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    Hello, wonderboy1953!

    You are given: . \begin{array}{ccccc}(1) & a + b + c + d &=&  e + f + g + h\\<br />
(2) &a + 2b + 3c + 4d &=& e + 2f + 3g + 4h \end{array}

    Now give me back: . (3)\;\; 4a + 3b + 2c + d \;=\; 4e + 3f + 2g + h

    Multiply [1] by 5: . 5a + 5b + 5c + 5d \;=\;5e + 5f + 5g + 5h

    . . . . . . (a + 4a) + (2b + 3b) + (4d + d) \;=\;(e+4e) + (2f + 3f) + (3g+2g) + (4h+h)

    . . . \underbrace{(a+2b + 3c + 4d)} + (4a \;+\; 3b \;+\; 2c \;+\; d) \;=\;\underbrace{(e+2f + 3g + 4h)} + (4e \;+\; 3f \;+\; 2g \;+\; h)
    . . . . . . . . . . . . . \nwarrow\quad\text{These are equal}\quad\nearrow


    Therefore: . 4a + 3b + 2c + d \;=\;4e + 3f + 2g + h

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  3. #3
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    You have proven it, Soroban

    Now what is needed are the numbers for a,b,c,d,e,f,g,h to complete the monograde equation that is being dot multiplied (hint: there can be more than one set of numbers to plug in).
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  4. #4
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    Quote Originally Posted by wonderboy1953 View Post
    Now give me back:
    (3) 4a + 3b + 2c + d = 4e + 3f + 2g + h
    OK; giving it to you BACK:
    h + g2 + f3 + e4 = d + c2 + b3 + a4 (3)

    You're right...easy...
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  5. #5
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    Extending my puzzle

    Soroban did a wonderful job with my puzzle (and in record time). Now for my next related puzzle.

    I give you a trigrade:

    a^n + b^n + c^n + d^n = e^n + f^n + g^n + h^n; n = 1,2,3 (although not required, for simplicity's sake, let a through h be positive integers).

    Now I also give you a + 2^2b + 3^2c +4^2d = e + 2^2f + 3^2g + 4^2h

    Can you give me 4^2a + 3^2b + 2^2c + d = 4^2e + 3^2f + 2^2g + h?

    (I see no one as yet assigned values to the letters so check out 2,8,9,15 = 3,5,12,14; also check out 1,8,10,17 = 2,5,13,16 when you substitute for the letters in my first and second puzzles).

    I'll be back next Sunday to check on your progress with my puzzles.
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  6. #6
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    I do have a solution

    I'm getting more familiarized with LaTex, Sigma notation, etc to compact the solution and other reasons. I hope to post it in this week.

    I'll have some more puzzles down the road.

    Happy New Year.
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