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Math Help - Vectors

  1. #1
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    Vectors

    If a = 3x+2y and b=5x-4y, find x and y in terms of a and b.

    Note: bolded=vectors (arrows on top)

    So this is my attempt, but leads me to no enlightenment.. Please demonstrate to me the method to proceed to the correct answer.

    x = (a/3) - (2y/3)
    y = (-b/4) +(5x/4)
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  2. #2
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    Quote Originally Posted by skeske1234 View Post
    If a = 3x+2y and b=5x-4y, find x and y in terms of a and b.

    Note: bolded=vectors (arrows on top)

    So this is my attempt, but leads me to no enlightenment.. Please demonstrate to me the method to proceed to the correct answer.

    x = (a/3) - (2y/3)
    y = (-b/4) +(5x/4)
    2\vec{a} = 6\vec{x} + 4\vec{y}

    \vec{b} = 5\vec{x} - 4\vec{y}

    ------------------------------------------------ add em' up ...

    2\vec{a} + \vec{b} = 11\vec{x}

    solve for vector x ...

    \vec{x} = \frac{2}{11} \vec{a} + \frac{1}{11} \vec{b}
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  3. #3
    Member Mauritzvdworm's Avatar
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    equation 1: \overline{a}=3\overline{x}+2\overline{y}
    equation 2: \overline{b}=5\overline{x}-4\overline{y}

    multiply equation 1 by 2 and add the result to the second equation

    2\overline{a}+\overline{b}=11\overline{x}
    hence, solving for x we obtain
    \overline{x}=\frac{1}{11}(2\overline{a}+\overline{  b})

    to solve for \overline{y} we need to multiply the first equation by 5 and the second by 3 then subtract the second from the first, we obtain

    5\overline{a}-3\overline{b}=22\overline{y}
    solving for y we find
    \overline{y}=\frac{1}{22}(5\overline{a}-3\overline{b})
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  4. #4
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    Quote Originally Posted by skeske1234 View Post
    If a = 3x+2y and b=5x-4y, find x and y in terms of a and b.
    We can write this in matrix form: \left[ \begin{gathered}<br />
  a \hfill \\<br />
  b \hfill \\ <br />
\end{gathered}  \right] = \left[ {\begin{array}{rr}<br />
   3 & 2  \\<br />
   5 & { - 4}  \\<br />
 \end{array} } \right]\left[ \begin{gathered}<br />
  x \hfill \\<br />
  y \hfill \\ <br />
\end{gathered}  \right]

    So \left[ \begin{gathered}<br />
  x \hfill \\<br />
  y \hfill \\ <br />
\end{gathered}  \right] = \left[ {\begin{array}{rr}<br />
   {\frac{2}<br />
{{11}}} & {\frac{1}<br />
{{11}}}  \\<br />
   {\frac{5}<br />
{{22}}} & {\frac{{ - 3}}<br />
{{22}}}  \\<br />
 \end{array} } \right]\left[ \begin{gathered}<br />
  a \hfill \\<br />
  b \hfill \\ <br />
\end{gathered}  \right]
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