urlacainnxp

2022-01-30

Finding an expression for x and y variables in terms of the corresponding image X and Y variables, find the image of the line $y=2x-7$ under the transformation given by
$M=\left(\begin{array}{cc}1& -3\\ -3& 2\end{array}\right)$

trovabile4p

Step 1
$\left(\begin{array}{ccc}1& & -3\\ -3& & 2\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}x-3y\\ -3x+2y\end{array}\right)=\left(\begin{array}{c}X\\ Y\end{array}\right).$
So, if
$y=2x-7$,
then
$X=x-3y=-5x+21$
and
$Y=-3x+2y=x-14$

shimmertulipsog

Step 1
Write the equation in matrix form:
$\left(\begin{array}{cc}2& -1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=7.$
Step 2
Modify the equation so it includes
$\left(\begin{array}{c}X\\ Y\end{array}\right)=\left(\begin{array}{cc}1& -3\\ -3& 2\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$
$\left[\left(\begin{array}{cc}2& -1\end{array}\right){\left(\begin{array}{cc}1& -3\\ -3& 2\end{array}\right)}^{-1}\right]\left[\left(\begin{array}{cc}1& -3\\ -3& 2\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\right]=7$
Step 3
Simplify:
$\left(\begin{array}{cc}-\frac{1}{7}& -\frac{5}{7}\end{array}\right)\left(\begin{array}{c}X\\ Y\end{array}\right)=7.$
So we have $X+5Y=-49$

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