Given the linear transformation \( T:\mathbb{R}^2 \rightarrow \mathbb{R}^2 \) defined by \( T(x, y) = (2x+y, x+3y) \). Find the pre-image of the point (5, 11) under the transformation T.
Answer
Substitute \(y = \frac{17}{5}\) into the first equation to solve for x: \[2x + \frac{17}{5} = 5 \implies 2x = \frac{8}{5} \implies x = \frac{4}{5} \]
Step 1 ：First, set the image of T equal to the given point (5, 11), which gives us the system of equations: \[ \begin{cases} 2x+y = 5 \\ x+3y = 11 \end{cases} \]
Step 2 ：Next, solve the system of equations. Multiply the second equation by 2 and subtract the first equation from the result to eliminate x. This gives us: \[ 2x+6y-2x-y = 22-5 \implies 5y = 17 \implies y = \frac{17}{5} \]
Step 3 ：Substitute \(y = \frac{17}{5}\) into the first equation to solve for x: \[2x + \frac{17}{5} = 5 \implies 2x = \frac{8}{5} \implies x = \frac{4}{5} \]
