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Math Help - Another matrix problem

  1. #1
    Junior Member
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    Post Another matrix problem

    Hi I'm stuck in this problem again.
    Can somebody please help me? Thank you.

    Using Gaussian elimination with back-substition or Gauss-Jordan Elimination solve the following problem.

    A small corporation borrowed $500,000 to expand its product line. Some of the money was borrowed at 9%, some at 10%, and some at 12%. How much was borrowed at each rate if the annual interest was $52,000 and the amount borrowed at 10% was 2.5 times the amount borrowed at 9%?
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  2. #2
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    Hello, cuteisa89!

    I'll set up for you . . .


    Using Gaussian elimination, solve the following problem.

    A small corporation borrowed $500,000 to expand its product line.
    Some of the money was borrowed at 9%, some at 10%, and some at 12%.
    How much was borrowed at each rate if the annual interest was $52,000
    and the amount borrowed at 10% was 2.5 times the amount borrowed at 9%?

    Let: . \begin{array}{c}x = \text{amount borrowed at 9\%} \\ y = \text{amount borrowed at 10\%} \\ z = \text{amount borrowed at 12\%}\end{array}

    The total borrowed is $500,000: . x + y + z \:=\:500,000 .(a)

    The interest on x dollars at 9% is: 0.09x dollars.
    The interest on y dollars at 10% is: 0.10y dollars.
    The interest on z dollars at 12% is: 0.12z dollars.
    . . The total interest was $52,000: . 0.09x + 0.10y + 0.12z \:=\:52,000
    Multiply by 100: . 9x + 10y + 12z \:=\:5,200,000 .(b)

    The amount borrowed at 10% was 2.5 times the amount borrowed at 9%:
    . . y \:=\:2.5x\quad\Rightarrow\quad 2.5x - y \:=\:0
    Multiply by 2: . 5x - 2y \:=\:0 .(c)

    And we have the sytem: . \begin{vmatrix}1 & 1 & 1 & | & 50,000 \\ 9 & 10 & 12 & | & 5,200,000 \\ 5 & \text{-}2 & 0 & | & 0\end{vmatrix}

    Now solve it . . .

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  3. #3
    Junior Member
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    Mar 2007
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    Smile

    So would it be:
    x= 100,000
    y = 250,000
    z = 150,000

    Is this correct?
    By the way, thank you so much for helping me! Really appreciated it!
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