# The transformation <mtext mathvariant="bold">T</mtext> maps points ( x , y ) of the

The transformation $\mathbf{\text{T}}$ maps points $\left(x,y\right)$ of the plane into image points $\left({x}^{\prime },{y}^{\prime }\right)$ such that
$\begin{array}{rl}{x}^{\prime }& =4x+2y+14\\ {y}^{\prime }& =2x+7y+42\end{array}$
Find the coordinates of the invariant point of $\mathbf{\text{T}}$. Hence express $\mathbf{\text{T}}$ in the form
$\left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }+k\end{array}\right)=\mathbf{\text{A}}\left(\begin{array}{c}x\\ y+k\end{array}\right)$
where $k$ is a positive integer and $\mathbf{\text{A}}$ is a $2×2$ matrix.
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Allison Pena
A transformation of the form $A\left[\begin{array}{c}x\\ y\end{array}\right]$ is linear: it maps the origin to the origin; that's not what your $T$ does. For an affine transformation, you need to add the constant term. So your $T$ is of the form
$T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}4& 2\\ 2& 7\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}14\\ 42\end{array}\right].$
So you have
$Tv=Av+r.$
A fixed point will satisfy
$Av+r=v,$
which we may rewrite as
$\left(A-I\right)v=-r.$
So that's the linear system you are looking for:
$\left[\begin{array}{cc}4-1& 2\\ 2& 7-1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=-\left[\begin{array}{c}14\\ 42\end{array}\right].$