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Math Help - Systems of Equations/Inequalities

  1. #1
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    Question Systems of Equations/Inequalities

    1.) Solve the following system of equations using Gaussian elimination:
    3w+10x-2y+3z=55
    w+12x-11y-z= 69
    -3w-6x-5y-10z=-47
    -3w+9x+5y+4z=-1

    Here's what I did so far:
    swap R1 and R2 (row 1 and row 2)
    -3R1+R2 (multiply row 1 by -3 and add row 2 to that product)
    3R1+R3
    3R1+R4

    Now, after that I think I may have made a mistake because I eventually get some very odd fractions that could not possibly be correct. Can somebody tell me what the next steps are???

    2.) Rachel has 3 investments for a total of $9000. They earn interest at 5%, 7%, and 9%, respectively. The total income is $7100. The income from the 9% investment exceeds the total from the other two investments by $1900. How much money is the investment that earns the 5% interest?

    I have absolutely no clue how to write the system for this. Can somebody at least help me get started???

    I would really appreciate your help on this! Thank you!!!
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  2. #2
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    Hello, \text{I}\heartsuit\text{themusic29} !

    I assume there is a typo in #2 . . .


    2.) Rachel has 3 investments for a total of $90,000.
    They earn interest at 5%, 7%, and 9%, respectively.
    The total income is $7100.
    The income from the 9% investment exceeds the total from the other two investments by $1900.
    How much money is the investment that earns the 5% interest?

    Let . \begin{array}{ccc}x &=& \text{amount at 5\%} \\ y &=& \text{amount at 7\%} \\ z &=& \text{amount at 9\%} \end{array}

    A total of $90,000 was invested: . x + y + z \:=\:90,000\;\;{\color{blue}[1]}

    The total interest was $7100.
    . . 0.05x + 0.07y + 0.09z \:=\:7100 \quad\Rightarrow\quad 5x + 7y + 9z \:=\:710,000\;\;{\color{blue}[2]}

    The interest from 9% is 1900 more than the total of the 5% and 7% investments.
    . . 0.09z \:=\:0.05x + 0.07y + 1900\quad\Rightarrow\quad 5x + 7y - 9z \:=\:-190,000 \;\;{\color{blue}[3]}

    Subtract [3] from [2]: . 18z \:=\:900,000 \quad\Rightarrow\quad z \:=\:50,000


    \begin{array}{ccccccccc}\text{Substitute into {\color{blue}[1]}:} & x + y + 50,000 &=&90,000 & \Rightarrow & x + y &=& 40,000 & {\color{blue}[3]}\\<br />
\text{Substitute into {\color{blue}[2]}:} & 5x + 7y + 450,000 &=& 710,000 & \Rightarrow & 5x + 7y &=& 260,000 & {\color{blue}[4]}<br />
\end{array}

    \begin{array}{cccc}\text{Multiply {\color{blue}[3]} by 7:} & 7x + 7y &=& 280,000 \\ \text{Subtract {\color{blue}[4]}:} & 5x + 7y &=& 260,000 \end{array}

    And we get: . 2x \:=\:20,000\quad\Rightarrow\quad x \;=\;10,000


    Therefore, $10,000 was invested at 5%.

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  3. #3
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    Quote Originally Posted by iheartthemusic29 View Post
    1.) Solve the following system of equations using Gaussian elimination:
    3w+10x-2y+3z=55
    w+12x-11y-z= 69
    -3w-6x-5y-10z=-47
    -3w+9x+5y+4z=-1

    Here's what I did so far:
    swap R1 and R2 (row 1 and row 2)
    -3R1+R2 (multiply row 1 by -3 and add row 2 to that product)
    3R1+R3
    3R1+R4

    Now, after that I think I may have made a mistake because I eventually get some very odd fractions that could not possibly be correct. Can somebody tell me what the next steps are???

    2.) Rachel has 3 investments for a total of $9000. They earn interest at 5%, 7%, and 9%, respectively. The total income is $7100. The income from the 9% investment exceeds the total from the other two investments by $1900. How much money is the investment that earns the 5% interest?

    I have absolutely no clue how to write the system for this. Can somebody at least help me get started???

    I would really appreciate your help on this! Thank you!!!
    Swap r1 and R2
    -3r1+r2
    3r1+r3
    3r1+r4
    26/12r1+r2
    -30/12r1+r3
    These are the next few steps to number 1. Is this what you did?
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