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Math Help - 9 ?'s that i really need help on!!!

  1. #1
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    9 ?'s that i really need help on!!!

    questions are on attachment!
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  2. #2
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    #1:

    Let x=amount invested at 10%.

    Then 3500-x is the amount invested at 15%.

    The difference between the two interest rates is $85.

    .15(3500-x)-.10x=85

    Also, how about showing some of your attempts instead of just posting a littany of problems.
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  3. #3
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    Hello, Lane!

    Here are a few more . . .


    Solve: . \begin{array}{cc}3x - 4y \,= \,7 \\ x + 2y \,=\,9\end{array}

    With no method specified, I'll use Elimination.

    We have: . \begin{array}{cc}3x - 4y \:= \:7 \\ x + 2y \:=\:9\end{array}\;\begin{array}{cc}(1)\\(2)\end{a  rray}

    Multiply (2) by 2: . 2x + 4y \:=\:18
    . . . . . . Add (1): . 3x - 4y \:=\:7

    And we have: . 5x\,=\,25\quad\Rightarrow\quad \boxed{x = 5}

    Substitute into (2): . 5 + 2y\:=\:9\quad\Rightarrow\quad 2y \,=\,4\qiad\Rightarrow\quad \boxed{y = 2}



    Solve with Cramer's Rule: . \begin{array}{cc}6x - 4y \:=\:6 \\ 3x \:= \:2y - 5\end{array}

    We have: . \begin{array}{cc}6x-4y\:=\:7\\3x-2y\:=\:\text{-}5\end{array}

    D\:=\:\begin{vmatrix}6 & \text{-}4 \\ 3 & \text{-}2\end{vmatrix} \;= \;(6)(\text{-}2) - (\text{-}4)(3) \;= \;-12 + 12 \;= \;0 . . . No solution!



    6. Solve: . \begin{array}{cc}xy\:=\:10 \\ x^2 + 4y^2\:=\:36\end{array}\;\begin{array}{cc}(1)\\(2) \end{array}

    Another trick question!

    From (1), we have: . y = \frac{10}{x}

    Substitute into (2): . x^2 + 4\left(\frac{10}{x}\right)^2\;=\;36 \quad\Rightarrow\quad x^2 + \frac{400}{x^2}\:=\:36

    Multiply by x^2:\;\;x^4 + 400 \:=\:36x^2\quad\Rightarrow\quad x^4 - 36x^2 + 400 \:= \:0

    Quadratic Formula: . x^2\:=\:\frac{-(\text{-}36) \pm \sqrt{(\text{-}36)^2 - 4(1)(400)}}{2(1)}

    . . . x^2\;=\;\frac{36 \pm\sqrt{-304}}{2} . . . No real root


    Therefore, the system has no real solution.

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