Question

The image of the point (2, -4) under the transformations $(x,y)\to (x+3y,y-x)$ is $(a,b)$ then $3b-2a$ is equal to

Answer 1

Let $(x_1,y_1)$ be the image of the point (x, y). Then by the given transformation, we have
$x_1=1\times x+3\times y$
$y_1=(-1)x+1\times y$
$\Rightarrow \left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]=\left[\begin{array}{cc}1& 3\\ -1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}1& 3\\ -1& 1\end{array}\right]\left[\begin{array}{c}2\\ -4\end{array}\right]=\left[\begin{array}{c}-10\\ -6\end{array}\right]$
Therefore, the image is $(-10,-6).$