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Math Help - Matrices!

  1. #1
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    Matrices!

    I'm not familiar with LaTex, so bear with my notation (I apologize in advance)...
    Find the values of x and y for which...
    [2y+5 = [x-1
    y-2] = 3x]
    is true.
    My work thus far
    2y+5=x-1; (0,-3)
    y-2=x-1; (-2/3, 0)
    ...I just don't know what steps to take particularly.
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  2. #2
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    Hello, puzzledwithpolynomials!

    Solve for x\text{ and }y\!:\;\;\begin{bmatrix}2y+5 \\ y-2\end{bmatrix} \:=\:\begin{bmatrix}x-1 \\ 3x \end{bmatrix}

    This can be written: . \begin{bmatrix}x-2y \\ 3x-y\end{bmatrix} \:=\:\begin{bmatrix}6\\\text{-}2\end{bmatrix}

    We have: . \left[\begin{array}{cc|c}1 & \text{-}2 & 6 \\ 3 & \text{-}1 & \text{-}2 \end{array}\right]

    . \begin{array}{c} \\ R_3-3R_1 \end{array} \left[\begin{array}{cc|c}1 & \text{-}2 & 6 \\ 0 & 5 & \text{-}20 \end{array}\right]

    . . . . \begin{array}{c} \\ \frac{1}{5}R_2 \end{array} \left[\begin{array}{cc|c}1 & \text{-}2 & 6 \\ 0 & 1 & \text{-}4 \end{array}\right]

    . \begin{array}{c}R_1+2R_2 \\ \\ \end{array} \left[\begin{array}{cc|c}1 & 0 & \text{-}2 \\ 0 & 1 & \text{-}4 \end{array}\right]


    Therefore: . \begin{Bmatrix}x &=& \text{-}2 \\ y &=& \text{-}4 \end{Bmatrix}

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  3. #3
    MHF Contributor
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    Talking

    Quote Originally Posted by puzzledwithpolynomials View Post
    I'm not familiar with LaTex, so bear with my notation (I apologize in advance)...
    Find the values of x and y for which...
    [2y+5 = [x-1
    y-2] = 3x]
    is true.
    If the previous reply's interpretation is correct, you have the following matrix equality:

    . . . . . \left[\begin{array}{c}2y\, +\, 5\\y\, -\, 2\end{array}\right]\, =\, \left[\begin{array}{c}x\, -\, 1\\3x\end{array}\right]

    It may be simpler, in this case, to use what you know about matrix equality to create regular equations:

    . . . . . 2y\, +\, 5\, =\, x\, -\, 1
    . . . . . y\, -\, 2\, =\, 3x

    Rearrange to get the variables on one side of the "equals" signs, and the constant terms on the other:

    . . . . . -x\, +\, 2y\, =\, -1\, -\, 5
    . . . . . -3x\, +\, y\, =\, +2

    . . . . . -1x\, +\, 2y\, =\, -6
    . . . . . -3x\, +\, 1y\, =\, 2

    Multiply the first row by -3 and add down to get:

    . . . . . .. 3x\, -\, 6y\, =\, 18
    . . . . . \underline{-3x\, +\, 1y\, =\, \mbox{ }\, 2}

    . . . . . . . . .. -5y\, =\, 20

    Etc, etc.
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