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Question Number 57805 by pete last updated on 12/Apr/19

$$\mathrm{Find}\mathrm{the}\mathrm{image}\mathrm{of}\mathrm{y}=3\mathrm{x}+1\mathrm{under}\mathrm{the}$$$$\mathrm{mapping}\left(\begin{array}{c}23\\ 12\end{array}\right).$$

Commented by pete last updated on 12/Apr/19

$$\mathrm{thanks}\mathrm{very}\mathrm{much}\mathrm{sir}$$

Commented by mr W last updated on 12/Apr/19

$$point1(0,1)$$$$\Rightarrow image(3,2)$$$$point2(-\frac{1}{3},0)$$$$\Rightarrow image(-\frac{2}{3},-\frac{1}{3})$$$$eqn.ofimage:$$$$\frac{y+\frac{1}{3}}{x+\frac{2}{3}}=\frac{2+\frac{1}{3}}{3+\frac{2}{3}}$$$$\frac{3y+1}{3x+2}=\frac{7}{11}$$$$\Rightarrow 11y=7x+1$$

Answered by mr W last updated on 13/Apr/19

$$letimageofpoint(x,y)be(u,v),$$$$\left(\begin{array}{c}u\\ v\end{array}\right)=\left(\begin{array}{cc}2& 3\\ 1& 2\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$$$$\Rightarrow \left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{cc}2& -3\\ -1& 2\end{array}\right)\left(\begin{array}{c}u\\ v\end{array}\right)$$$$\Rightarrow x=2u-3v$$$$\Rightarrow y=-u+2v$$$$\Rightarrow -u+2v=3(2u-3v)+1$$$$\Rightarrow 11v=7u+1$$$$i.e.imageis11y=7x+1$$

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