1. 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)

2. Originally Posted by skeske1234
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)
$\displaystyle 2\vec{a} = 6\vec{x} + 4\vec{y}$

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

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

solve for vector x ...

$\displaystyle \vec{x} = \frac{2}{11} \vec{a} + \frac{1}{11} \vec{b}$

3. equation 1: $\displaystyle \overline{a}=3\overline{x}+2\overline{y}$
equation 2: $\displaystyle \overline{b}=5\overline{x}-4\overline{y}$

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

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

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

$\displaystyle 5\overline{a}-3\overline{b}=22\overline{y}$
solving for y we find
$\displaystyle \overline{y}=\frac{1}{22}(5\overline{a}-3\overline{b})$

4. Originally Posted by skeske1234
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: $\displaystyle \left[ \begin{gathered} a \hfill \\ b \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{rr} 3 & 2 \\ 5 & { - 4} \\ \end{array} } \right]\left[ \begin{gathered} x \hfill \\ y \hfill \\ \end{gathered} \right]$

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